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qq^ 74l2' q17167S GENERAL THEORY OF CONICAL FLOWS AND ITS APPLICATION TO SUPERSONIC AERODYNAMICS By Paul Germain Preface By M. J. Peres NOTICE This report deals with a method of studying the equation of cylin drical waves particularly indicated for the solution of certain problems in aerodynamics. One of the most remarkable aspects of this method is that it reduces problems of a hyperbolic equation to problems of harmonic functions. We have applied ourselves here to setting up the fundamental principles, to developing their investigation up to calculation of the pressures on the visualized obstacles, and to showing how the initial field of "conical flows" was considerably enlarged by a procedure of integral superposition. Such an undertaking entails certain dangers. In France the exist ence of conical flows was not known before 1946. Abroad, this question has, for a long time, given rise to numerous reports which either were not published or were published only after a certain delay. Thus it must be pointed out that some of the results here obtained, original in France when found, doubtlessly were not original abroad. Nevertheless it seems possible to me to specify a certain number of points treated in this report which, even considering the lapse of time, appear as new: the parts concerning homogeneous flows, the general study of conical flows with infinitesimal cone angles, the numerical or analogous methods for the study of flows flattened in one direction, and a certain number of the results of chapter IV. Moreover, even where the results which we found independently were already known abroad, the employed methods are not always identical. Another peculiarity should be noted. Since these questions actually are everywhere the object of numerous investigations, progress has made very rapid strides. This report edited at the beginning of 1948, risks appearing, in certain aspects, slightly outmoded in 1949. To extenuate this inconvenience we have indicated in a brief appendix placed at the end of this report the progress made in these questions during the last year. This appendix is followed by a supplementary bibliography which indicates recent reports concerning our subject, or older ones of which we had no previous knowledge. I should not have been able to successfully terminate this report without the advice and support of my teacher, Mr. J. Peres, and it is very important to me to express here my great respect for and gratitude to him. I should equally cite all those who directly or less directly have contributed to my intellectual development and to whom I owe so much: my teachers of special mathematics and of normal school, Mr. Bouligand who directed my first reports, Mr. Villat, promoter of the Study of the Mechanics of Fluids in France whose brilliant instruction has been of the greatest value to me. I also feel obliged to thank the directors of the O.N.E.R.A. who have facilitated my task, and especially Mr. Girerd, director of aero dynamic research. PREFACE With his research on conical flows and their application, Mr. Paul Germain has made a major contribution to the very timely study of super sonic aerodynamics. The present volume offers a comprehensive expose which had been still lacking, an expose of elegance and solid construc tion containing a number of original developments. The author has fur thermore considered very thoroughly the applications and has shown how one may solve within the scope of linear theory, by combinations of conical flows, the general problems of the supersonic wing, taking into account dihedral and sweepback, and also fuselage and control surface effects. The analysis he develops in this respect leads him to methods which permit, either by calculation alone or with the support of electrolytictank experimentation, complete and accurate numerical determinations. After a few preliminary developments (particularly on the validity of the hypothesis of linearization), chapter I is devoted to the gener alities concerning conical flows. In such flows the velocity components depend only on two variables and their determination makes use of har monic functions or of functions which verify the wave equation with two variables according to whether one is inside or outside of the Mach cone. Mr. Germain specifies the conditions of agreement between func tions defined in one domain or in the other and shows that the study of conical flows amounts in general to boundary problems relative to three analytical functions connected by differential relationships. He studies, on the other hand, homogeneous flows which generalize the cone flows and are no less useful in the applications. From the viewpoint of the linear theory of supersonic flows one must maintain two principal types of conical flows, bounded respectively by an obstacle in the form of a cone with infinitesimal cone angle, and by an obstacle in the form of a cone flattened in one direction. The general investigation of the flows of the first type is entirely Mr. Germain's own and forms the object of chapter II of his book. By a subtle analysis of the approximations which may be legitimate Mr. Germain succeeds in simplifying the rather complex boundary problem he had to deal with; he replaces it by an external Hilbert problem. He shows how it is possible, after having obtained the solution for an orientation of the cone in the relative air stream, to pass, in a manner as simple as it is elegant, to the calculation of the effect of a change in inci dence. He gives general formulas for the forces, treats completely diverse noteworthy special cases and finally applies the method of trigo nometric operators which is also his own to the practical numerical calculation of the flow about an arbitrary cone. The determination of movements about infinitely flattened cones has formed the object of numerous reports. The analysis which Mr. Germain develops for this question (chapter III) contributes simplifications, specifications, and important supplements. Thus he evolves, in the case of an obstacle inside the Mach cone, a principle of minimum singularity which enters into the determination of the solution. Mr. Germain gives two original methods for treatment of the general case: one utilizes the electrolytictank analogy, surmounting the difficulty arising from the experimental application of the principle of minimum singularity; the other, purely numerical, involves the trigonometric operators quoted above. In the last chapter, finally, Mr. Germain visualizes the composi tion of conical flows with regard to aerodynamic calculation of a super sonic aircraft. Concerning this subject he develops a complete theory which covers most of the known results and incorporates new ones. He concludes with an outline of the flows past a flat dihedral, with appli cation to the fins and control surfaces. The creation of the National Office for Aeronautical Study and Research has already made possible the setting up of groups of investi gators which do excellent work in several domains that are of interest to modern aviation and put us on the level of the best research centers abroad. Mr. Paul Germain inspirits and directs one of those groups in the most efficient manner. He is one of those, and the present report will suffice to bear out this statement, on whom we can count for the development of the study of aerodynamics in France. Joseph Peres Member of the Academy of Sciences NACA TM 1554 TABLE OF CONTENTS Pages CHAPTER I GENERALITIES ON CONICAL FLOWS . 1 1.1 Equations of Supersonic Linearized Flows ... 1 1.2 Generalities on Conical Flows . .. 10 1.5 Homogeneous Flows . . ... ... .22 CHAPTER II CONICAL FLOWS WITH INFINITESIMAL CONE ANGLES ..... 30 2.1 Solution of the Problem . ... 30 2.2 Applications . . ... 41 Cone of Revolution . . ... .44 Elliptic Cone ....... . .. ..... 47 Study of a Cone With Semicircular Section . .. .58 2.5 Numerical Calculation of Conical Flows With Infinitesimal Cone Angles . . .. .62 Calculation of the Trigonometric Operators ... 68 CHAPTER III CONICAL FLOWS INFINITELY FLATTENED IN ONE DIRECTION . . . ... 79 5.1 Cone Obstacle Entirely Inside the Mach Cone ... 80 Study of the Elementary Problems (Symmetrical Cone ..... 80 Flows With Respect to Oxlx)) . .. ..80 Nonsymmetrical Conical Flows . .97 General Problem . . ... ...... 105 RheoElectric Method . .... .108 Purely Numerical Method ................. 117 5.2 Case Where the Cone Is Not Inside the Mach Cone () 152 Cone Totally Bisecting the Mach Cone (Fig. 28) ... 154 Cone Partially Inside and Partially Outside of the Mach Cone (r) (Fig. 50) . . ... .. .142 Cone Entirely Outside of the Cone (C) (Fig. 29) . 152 5.5 Supplementary Remarks on the Infinitely Flattened Conical Flows . . ... 159 CHAPTER IV THE COMPOSITION OF CONICAL FLOWS AND ITS APPLICATION TO THE AERODYNAMIC CALCULATION OF SUPERSONIC AIRCRAFT .. 168 4.1 Calculation of the Wings . .. 168 Symmetrical Problems . . ... ... 171 Rectangular Wings . .... ..... 171 Sweptback Wings . . ... 186 Lifting problems . . ... ..... 206 Rectangular Wings . . ... 214 Effect of Ailerons and Flaps . ... .225 Sweptback Wing . . ... .. 229 The Lifting Segments . .... ... 240 4.2 Study of Fuselages . ... ... 243 4.5 Conical Flows Past a Flat Dihedral. Fins and Control Surfaces . . ... . 251 vi NACA TM 1554 Page REFERENCES .. .. .. . ...... 261 APPENDIX ................. ........... 264 Digitized by the Interne[ Archive in 2011 wilh funding Irom University of Florida, Geolge A. Smathers Libraries wilh support from LYRASIS and the Sloan Foundation Illp: www.arciiive.org details generaltheoryolc00unii NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM 1354 GENERAL THEORY OF CONICAL FLOWS AND ITS APPLICATION TO SUPERSONIC AERODYNAMICS* By Paul Germain CHAPTER I GENERALITIES ON CONICAL FLOWS 1.1 Equations of Supersonic Linearized Flows 1.1.1 General Equation for the Velocity Potential Let us visualize the permanent irrotational flow of a compressible perfect fluid for which the pressure p and the density p are mutual functions. The space in which the flow takes place will be fixed by three trirectangular axes Oxl, Ox2, Ox3, the coordinates of a fluid molecule will be xl, x2, x3, the projections on Oxi of the veloc ity V and of the acceleration A of a molecule will be denoted by ui and ai, respectively. The fundamental equations which permit determination are the Euler equations of the flow * 1  A grad p P ai 1 6P al P x, the equation of continuity *"La theorie ge6nrale des movements coniques et ses applications a l'aerodynamique supersonique." Office National d'Itudes et de Recherches Aeronautiques, no. 34, 1949. 1We employ the classic convention of the silent index: pui bxi v is to be read: ~(uI) + ~(Pu2) JL d S (Pu3) 6x NACA U' 1354 div pV = 0 or x(Pui) = 0 (1.2) and the equation of compressibility p = f(p) If one notes that 6ui ai = u 1 k 6xk and introduces the sonic velocity2 c2 dp dp the equation (I.1) assumes the form p1 i Sxi _ 1 dp 6p P dp 6xi (1.3) (1.4) _ 2 ap P 6xi We introduce the velocity potential (xl, x2, x3), defined with the exception of one constant, by V = grad 4 x  Ui ox, 2The velocity of sound, introduced here by the symbol d_ has a dp wellknown physical significance; it is the velocity of propagation of small disturbances. This significance frequently permits an intuitive interpretation of certain results which we shall encounter later on (see section 1.1.4). oui lk axk (1.5) NACA TM 1354 which is legitimate since we shall assume the flow to be irrotational. If we make the combination u.u i N D 620 Uiuk uk xi xk xi 0xk one sees, taking into account equations (1.5) and (1.2), that 6' 6c 620 26 24 63 2 c2 62 (1.6) 3xi Oxk 6xi 6xk 6x 2 i This equation is the general equation for the velocity potential. One may show, besides, that c is a function of the velocity modulus; thus one obtains an equation with partial derivatives of the second order, linear with respect to the second derivatives, but not completely linear. The nonlinear character of the equation for the velocity potential makes the rigorous investigation of compressible flows rather difficult, at least in the threedimensional case. In order to be able to study, at least approximately, the behavior of wings., fuselages, and other elements of aeronautical structures, at velocities due to the compressibility, one has been led to introduce simplifying hypothesis which permit "linearization" of the equation for the velocity potential. 1.1.2 The Hypotheses of Linearization and Their Consequences For aerodynamic calculation, one may assume that the body around which the flow occurs has a position fixed in space nd that the fluid at infinity upstream is moving with a velocity U, U being a constant vector, the modulus of which will be taken as velocity unit. We shall always assume that the axis Oxi has the same direction as U; the hypotheses of linearization amount to assuming that at every point of the fluid the velocity is reasonably equivalent to U. We put in a more precise manner S= 1+u u2 = v u3 = W NACA TM 1354 u, v, w are, according to definition, the components of the "pertur bation velocity." (1) u, v, w are quantities which are very small referred to unity; if one considers these quantities as infinitesimals of the first order, one makes it at least permissible to neglect3 in the equations all infinitesimals of the second order such as u2, v2, uv, etc. (2) All partial derivatives of u, v, w with respect to the coordinates are equally infinitesimals at least of the first order so that one is justified in neglecting terms such as u , etc. 0x1 k fx2 One may deduce from these hypotheses a few immediate consequences: (a) At every point of the field, the angle of the velocity vector with the axis Oxi is an infinitesimal of the first order at least. Hence there results a condition imposed on the body about which the flow is to be investigated; at every point the tangent plane must make a small angle with the direction of the nondisturbed flow (this is what one calls the uniform motion, defined by the velocity U). If one designates by q the velocity modulus, one has, taking the hypotheses setup into account q2 = (1 + u)2 + v2 + w2 = 1 + 2u whence q= 1 u (b) The pressure p and the density p differ from the values p, and pl which these magnitudes assume at infinity upstream only by an infinitesimal of the first order; the equation (1.5) is written in effect au C1 6xl P1 xl 3This signifies that u, v, w may very well not be infinitesimals of the same order; in this case one takes as the principal infinitesimal the perturbation velocity component which has the lowest order. NACA TM 1354 with cl denoting the sonic velocity at infinity upstream; thus 2  u 1 (P pl) 5 (1'7) On the other hand, according to equation (I.4) P Pl = c2 P) = lu If one defines the pressure coefficient Cp by SP P p l /2 12 one has Cp = 2u p (1.8) (c) Finally, an examination of what becomes of the equation for the velocity potential (equation (1.6)) under these hypotheses shows that it is reduced to x2 12 1 2 1 xl + _ x22 Let p(x1,x2,x3) be the "disturbance potential," that is, the potential the gradient of which is identical with the disturbance velocity vector; P (x,x2,x3) is the solution of the equation with partial derivatives of the second order 2 1 Cl2 _2 2 2 c1 6x1 _ i _ 2 dX2 (1.9) + 32 Ox,2 a completely linear equation. x \ 2 3 NACA fM 1354 The Mach number of the flow is called the dimensionless con stant M which, with the velocity unit to be chosen arbitrarily, cl is written here M = 1/c1. We put: (M2 1) = 02, with e being equal +1 or 1 according to whether M is larger or smaller than unity. (1) If M < 1, equation (I.9) is written 2 a29 2 29 P2 2 + 2p + 2 = 6x12 6x22 6x 2 an equation which may be easily reduced to the Laplace equation. This equation applies to flows called "subsonic" because the velocity of the nondisturbed flow is smaller than the sonic velocity at infinity upstream. These flows will not be investigated in the course of this report . (2) If M >1, equation (1.9) is thus written 2 a2, 2, +629 032 + (I.10) ax12 6x22 6x32 This equation applies to "supersonic" flows; if one interprets xl as representing the time t, this equation is identical with the equa tion for cylindrical waves, wellknown in mathematical physics. Investi gation of this equation will form the object of this report. Remarks. (1) It should be noted that, in order to write the preceding equa tion, it was not necessary to specify the form of the equation for the state of the fluid. In particular, the formulas written above do not introduce the value of the exponent 7 of the adiabatic relation p = kpy which is the form usually assumed by the equation of compressibility. Investigation of linear subsonic flows has formed the object of numerous reports. See references 1 and 2. NACA TM 1354 (2) The preceding analysis shows clearly the very different char acter of subsonic flows which lead to an elliptic equation, and of supersonic flows which are represented by a hyperbolic equation. (3) When we wrote equation (I.9), we supposed implicitly that M2 1 was not infinitely small, that is, that the flow was not "tran sonic," according to the expression of Von KArman5. Thus it is impossible to make M tend toward unity in the results we shall obtain, in the hope to acquire information on the transonic case6. (4) It may happen, in agreement with the statement made in foot note 3, that u is an infinitesimal of an order higher than first. In this case, one will take up again the analysis made in paragraph (b) of section 1.1.2, which leads to a formula yielding the Cp, more adequate than the formula (1.8) Cp = 2u (v2 + w2) (.11) 1.1.3 Validity of the Hypotheses of Linearization Any simplifying hypothesis leads necessarily to results different from those which one would obtain with a rigorous method. Nevertheless, it was shown in certain numerical investigations on profiles (two dimensional flows) where the rigorous method and the method of lineari zation were applied simultaneously that the approximation method provided a very good approximation for the calculation of forces. Besides, it is wellknown that the classic Prandtl equation for the investigation of 5Study of the transonic flows, with simplifying hypotheses analogous to those that have been made, requires a more compact analysis of the phenomena. It leads to a nonlinear equation, described for the first time by Oswatitsch and Wieghart (ref. 3). From it one may very easily deduce interesting relations of similitude for the transonic flows (ref. 4). One may find these relations also, in a very simple manner, by utilizing the hodograph plane. In a general manner, according to the values of M, one may be led to neglect certain terms in the final formulas found for the pressure coefficient Cp. This requires an evaluation, in every particular case, of the order of magnitude of the terms occurring in the formulas when M varies. In this report, we shall never enter into such a discussion. We shall limit ourselves voluntarily to the general formulas. An inter esting example of such a discussion may be found in the recent memorandum of E. Laitone (ref. 5). NACA TM 1354 wings of finite span in an incompressible fluid furnishes very acceptable results, and the Prandtl equation results from a linearization of the rigorous problem. It happens frequently, we shall have occasion several times to point it out, that the solution found for u, v, w will not satisfy the hypotheses of section 1.1.2 in certain regions (for example in the neigh borhood of a leading edge); eventually certain ones among these magni tudes could even become infinite. Under rigorous conditions such a solution should not be retained. Anyhow, if the regions where the hypotheses of linearization are not satisfied are "sufficiently small," it is permissible to assume that the expressions found for the forces (obtained by integration of the pres sures) will still'remain valid. This constitutes a justification a posteriori for the linearization method so frequently utilized in numerous aerodynamic problems7. Therefore, we shall not systematically discard the solutions found which will not wholly satisfy the hypotheses we set up. 1.1.4 Limiting Conditions. Existence Theorem Physically, the definition of sonic velocity leads to the rule which has been called the "rule of forbidden signals" (see footnote 2 of section 1.1.1) and which can be stated as follows: A disturbance in a uniform supersonic flow, of the velocity U produced at a point P, takes effect only inside of a halfcone of revolution of the axis U and of the apex halfangle a = Arc sin(l/M); (D cot a) a is called the Mach angle, the halfcone in question Mach aftercone at P." Correlatively, one may state that the condition of the fluid at a point M (pressure, velocity, etc.) depends only on the character of the disturbances produced in the nondisturbed flow at points situated inside of the "Mach forecone at M;" the Mach forecone at a point is obviously the symmetrical counterpart of the Mach aftercone with respect to its apex. If one wants to justify this rule from the mathematical viewpoint, one must start out from the formulas solving the problem of Cauchy and take into account the boundary conditions particular to the problem. Along the obstacle one must write that the velocity is tangent to the obstacle which gives the value dP/dn. Moreover, at infinity 7 For instance, in the investigation of vibratory motions of infin itely small amplitude about slender profiles. NACA TM 1354 upstream (xl = c) the first derivatives of ( must be zero, since 9 is, from the aerodynamic viewpoint, only determined to within a constant, it will be assumed zero. The characteristic surfaces of the equation (I.10) are the Mach cones. If one of the Mach cones of the point P cuts off a region (R) on a surface (Z), the classic study of the problem of Cauchy8 shows that the value of 9 at P is a continuous linear function of the values of 9 and of dcp/dn on R. Let us therefore consider a point M of a supersonic flow such that its forecone does not intersect the obstacle. We take as the surface E a plane xl = A, with A being of arbitrary magnitude. On E, rP and dP/dn, which are continuous functions, will be arbi trarily small. Consequently the value of c at M is zero. Thus one aspect of the rule of "forbidden signal" is justified. Let ds suppose that the forwardcone of M cuts off a region r(M) on the obstacle; on r(M), dc/dn is given by the boundary conditions; thus 91(M) is a linear function of the values of C on r(M). One sees therefore that, if one makes M tend toward a point MO of the obstacle, one will obtain a functional equation permitting the determination of P on the obstacle, at least in the case where the existence and uniqueness of the solution will be insured9. Consequently, ((M) depends only on the values of CP/dn in the region r(M); this justifies the fundamental result of the rule of "forbidden signals."10 1.1.5 General Methods for Investigation of Linearized Supersonic Flows In a recent articlell dealing with the study of linear supersonic flows, Von Kirman indicates that two major general procedures exist for 8For the problem of Cauchy, relative to the equation for cylindrical waves, see for instance references 6 and 7. 9Such a method has been utilized by G. Temple and H. A. Jahn, in their study of a partial differential equation with two variables (ref. 8). 10A more exact investigation of this question may be found in appendix 1, at the end of this report. 11See reference 4. A quick expose of the methods in question may also be found in the text, in reference 2. NACA TM 1354 the study of these flows, one called "the source method," the other "the acoustic analogy." The first is an old method and its theoretical application is fairly simple. It consists in placing on the outer surface of the obstacle a continuous distribution of singularities, called sources, the superposition of which gives at every point of the space the desired potential; the local strength of the sources may, in general, easily be determined with the aid of the boundary conditions. The second method utilizes a fundamental solution of the equation (I.10), the composition of which permits one to obtain the desired potential; this procedure is interesting in that it permits utilization of the Fourier integrals and thus furnishes, at least in certain particular cases, rather simple expressions for the total energy. Von Karman also indicates, at the end of his report, a third general procedure, that of conical flows. We intend to investigate in this report the conical flows and the development of this third procedure which utilizes systematically the composition of the "conical flows" and, more generally, of the flows which we shall call "homogeneous flows of the order n." We shall see that this procedure permits one to find very easily, and frequently with less expenditure, a great number of the results previously obtained by other methods, and to bring to a successful end the investigation of certain problems which, to our knowledge, have not yet been solved. 1.2 Generalities on Conical Flows 1.2.1 History and Definition Conical flows have been introduced by A. Busemann (ref. 9) who has given the principal characteristics of these flows and has indicated briefly in what ways they could be utilized in the investigation of supersonic flows. Busemann gives as examples some results, frequently without proof. Several authors have supplemented the investigation of Busemann: Stewart (ref. 10) has studied the case of the lifting wing A to which we shall come back later on; L. Beschkine (ref. 11) has fur nished a certain number of results but generally without demonstration. We thought it of interest to attempt a summary of the entire problem. One calls "conical flows" (more precisely, "infinitesimal conical flows")12 the flows in which there exists a point 0 such that along 12The adjective "infinitesimal" is remindful of the fact that the flows have been linearizedd;" we shall henceforward omit this qualifica tion since no confusion can arise in this report. NACA TM 1354 every straight line issuing toward one side of 0, the velocity vector remains of the same value. Let (i) be a plane not containing 0, normal to the vector U; let us suppose only that the velocity vector at every point of (i) is not normal to (r); the projection of these velocity vectors on (i) determines a field of vectors, the lines of force of which we shall call (y): the cones (a) of vertex 0 and directrix (7) are "stream cones" for the flow. More generally, let (S) be a stream surface of the flow, passing through O; every surface deduced from (S) by homothety of the center 0 and of k, k being an arbitrary positive number, is a stream surface. (S) is not necessarily a conical surface of apex 0, but having (S) given as an obstacle does not permit one to foresee the existence of such a flow. It is different if a conical obstacle of apex 0 is given; the designation "conical flow" is thus justified. Conversely, let us consider a cone of the apex 0, situated entirely in the region xl >0, and suppose that a linearized supersonic flow exists around this cone; this flow is necessarily a conical flow such as has just been defined; in fact, if V(xlX2,x3) denotes this velocity field, V( xl,Xx2,Xxj) ( being any arbitrary positive number) is equally a velocity field satisfying all conditions of the problem; con sequently, if the uniqueness of the desired flow is admitted, 7 must be constant along every halfstraight line from 0 13. Let us also point out that according to equations (1.8) or (I.ll), the surfaces of equal pressure are also cones of the apex 0. 1.2.2 Partial Differential Equations Satisfied by the Velocity Components According to definition, the velocity components of a conical flow depend only on two variables; on the other hand, as functions of xl, 13It should be noted that this argument will no longer be valid without restriction in the case of a real supersonic flow around a cone because in this case the principle of "forbidden signals" is no longer valid in the rigorous form stated. Among other possibilities, a detached shock wave may form upstream from the cone behind which the motion is no longer irrotational. NACA TM 1354 x2, x3, they are naturally the solution of the equation S2f 6x3 2 O3 Let us first put x2 = r cos 9 x3 = r sin 0 the equation then assumes the form 2 32f 6xl2 2 2 _ 2f r1 2 1 rf or2 r2 602 r 3r (1.12) The second term of equation (1.12) is actually nothing else but the Laplacian of f(x1,x2,x3) in the plane x2, x3 (xl being con sidered as parameter); naturally f(xl,r,8) is periodic in 8, the period being equal to 2n. To make the conical character of the flow evident, let us put xI = prX (I.13) X is a new variable; X < 1 characterizes the exterior of the Mach cone with the apex 0, X > 1 characterizes the interior of the cone. Under these conditions, the disturbancevelocity components are func tions only of :. and e. Since f is a function of X and 0 only d2f = 2 dx2 dx2 + 232 dv dB + ax 6e 02f O82 + 6f d2X + kf d2 5ax e dX = r (dxl OX dr) o~r\ 1I d2x = L(d2x X d2r  pr 2dr dx 1 + 20 X dr2) r r 2 62f 6x 2 _ 2f bx22 NACA TM 1354 2f ;r2 62f 6e2 6f are the respective coefficients of dxl2, Tr dr2, d2r in the expression of d2f as a function of the vari s xl, r, e. As a consequence, the equation (1.12) becomes under these conditions r(x2 6A.O2 (1.14) + b2f + f = 0 +02  6e2 ax One may try to simplify this equation further by replacing the variable X by the variable t, X and t being connected by a rela tionship X = X()), and by making a judicious choice for the func tion x(S). The first operation gives (x2 2 i) 6t2 + y,2 r 602 + B 0 6 at (x. )x' = oS X' with the primes denoting derivatives with respect to E. this equation, one may make the term in disappear. realized by putting (1) If X >1, For simplifying This will be X = ch e one obtains for f Laplace's equation 62f 6t2 + 2 S62 (1.16) (2) If X < 1, = cos 1 (I.17) in this case, one obtains the equation for waves with two variables 62f 3,2 62f 682 (I.18) (1.15) NACA TM 1354 Geometrical interpretation. X > 1 corresponds to the interior of the Mach rearward cone (r) of the point 0; every semiinfinite line, issuing from 0, inside of this cone, has as image a point 8, E. One will assume, for instance, n < 0 < i; k = 0 corresponds to the cone (r), E = w corresponds to the cone axis (it will always be pos sible to assume as positive). The image of the interior of (r) forms therefore on the region (A) of the plane (0,a) (fig. 1), limited by the semiinfinite lines AT, A'T' and by the segment AA'. The correspondence is double valued in the sense that to a semiinfinite line issuing from 0 there corresponds one point and one only (0e,) in the bounded region and conversely, to one point of this region there corresponds one semiinfinite line, and one only, issuing from O, inside of (r). Since we shall suppose, in general, that the cone investigated is entirely in the region xl > 0, only this region will be of interest (P then being identically zero for xl < 0). The semiinfinite lines of this region issuing from 0, outside of (r), correspond to 0 < X < 1 (fig. 2), that is, according to equation (I.17), 0 < r < !L; n = 0 2 corresponds to the cone (F), T = n to the plane xI = 0; the semi 2 infinite lines issuing from 0 correspond biunivocally to the points of the region (A'), inside of the rectangle AA'B'B in the plane (e,r). Summing up, the velocity components satisfy the simple equa tions (I.16) and (I.18), the first of which is relative to the region (A), the second to the region (A'). 1.2.3 Fundamental Theorem The equation (1.14) which represents the fundamental equation of our problem is an equation of mixed type; it is elliptic or hyperbolic according to whether is larger or smaller than unity. In order to study this equation in a simpler manner, we have been led to divide the domain of the variables into two parts and to represent them on two different planes. How an agreement will be reached between the solutions obtained for f in the two planes that is the question which will be completely elucidated by the following theorem which will be fundamental in the course of our investigation. Theorem: There exists "agreement" as to X = 1 for all derivatives of f, defined in either the region (A) or (A'), provided that there is "agreement" for the function itself. NACA TM 1354 In fact, let us take two functions fl(e,), f2(6,Ti), the first satisfying the equation (1.16) in the region (A), the second the equa tion (I.18) in the region (A'), both assuming the same values c(e) on the respective segments ( 0 = 0, i < 0 < n) (r = 0, n < 0 < i). Let Inf B0 be the abcissa of a point of AA'. If 6nfL(80,0) exists, 8en Sd; consequently den nf 2 (0,0) en exists and ae oO aen a3f(e,, )  On Let us now pass to the investigation of the derivatives of the order n of the form nf the equation (1.14) shows first that 6Xoen1 f(e,1) = f(,1) OX 602 which proves that all partial derivatives of the order 1 with respect to X have the same value on (P), whether they are calculated starting from fl or from f2. The argument develops without difficulty through recurrence. By deriving equation (1.14) n times with respect to : and making X = 1, one obtains (2n + 1) + n2 + 6nf 0 An+1 .a 2" 2n which finally shows that the values n+f can be uniquely expressed ae. n as a function of the derivatives of f(0) with respect to G and that they, consequently, have the same value, whether calculated starting from fl or from f2. Summing up, one may say that it is sufficient for the establishment of the "agreement" between two solutions defined in (A) and (A'), if these solutions assume the same value on the segment AA'. anfl(00 agn 16 NACA TM 1354 1.2.4 Mode of Dependence of the SemiInfinite Lines Issuing From 0 If one puts in the plane (e0,) 8 + T = 2X 8 q = 2 (1.19) one sees that the characteristics of the equation (I.18) are the parallels to the bisectrices K = cte, = cte. These characteristics are, in the plane (1,e), the images of the planes xI = r cos(2A 0) and xl = or cos(O 2P) which are the planes tangent to the cone (P). The characteristics passing through a point s0(O0,0) are the images of two planes tangent to the cone (r) which one may lay through the semiinfinite A0 cor responding to the point 60 of the plane (8,0) (fig. 3). The gener atrices of contact are characterized on the cone by the values 81 and 82 of the angle 0. One encounters here a result which seems to contradict indications of section 1.1.4. This apparent contradiction is immediately explained if one notes that, since all points of a semi infinite A0 issued from 0 are equivalent, one must consider at the same time all Mach cones, the apexes of which are situated on A0; the group of these cones admits as envelope precisely the two planes tangent to the cone (r) passing through A0. We shall call "Mach dihedron posterior" to the semiinfinite A0 that one of the dihedra formed by the two planes which contains the group of the Mach cones to the rear of the points of A0. The region inside ot this dihedron and outside of the cone (r) has as image in the plane (0,rj) the triangle 01 5062. A semiinfinite A1 will be said to be dependent on or independent of A0 according to whether the image of A1 will be inside or outside of the triangle 81 6082. This argument also explains why the equa tion (1.14) shows elliptic character inside of (r). More precisely, two semiinfinite lines A1 and 62, inside of (r), are in a state of neutral dependence (ref. 9). In fact, let M1 be a point of Ak, M2 a point of 2; let us suppose that Mi is outside of the Mach forward cone of M2; according to the argument of section 1.1.4 the point M2 seems to be independent of Ml; but on the other hand, if one assumes Mi' NACA TM 1354 to be a point of A1, inside of the Mach forward cone of M2, M1' and M1 are equivalent which explains that M2 is actually not inde pendent of M1 (fig. 4). 1.2.5 The Conditions of Compatibility Thus one may foresee how the solution of a problem of conical flow will unfold itself. One will attempt to solve this problem in the region (A') which will generally be fairly easy since the general solu tion of the equation (1.18) is written immediately by adjoining an arbi trary function of the variable 6 + n to an arbitrary function of the variable 6 q. This will have the effect of "transporting" onto the segment AA' the boundary conditions relative to the region (A'). Applying the fundamental theorem, one will be led to a problem of har monic functions in the region (A). But taking as unknown functions the components u, v, w, of the disturbance velocity, we have introduced three unknown functions (while there was only one when we dealt with the function C). One must therefore write certain relationships of compatibility which express finally that the motion is indeed irrotational. The motion will be irrotational if u dxI + v dx2 + w dx3 is an exact differential which will be the case when, and only when x1 du + x2 dv + x dw = r(py du + cos 8 dv + sin 0 dw) is an exact differential. This can occur only if this expression is identically zero, with u, v, w being functions uniquely of 0 and of X (the total differential not containing a term.in dr must be independent of r): In a conical flow the potential is written P = uxl + vx2 + wx3 = r(puX + v cos 0 + w sin e) with u, v, w being the disturbancevelocity components. One will note that C, is proportional to r. Moreover OX du + cos 0 dv + sin 0 dw = 0 (1.20) 18 NACA TM 1354 This is the relationship which is to be written, and this is the point in question, on one hand in the plane (C,0), on the other in the plane (6,). (a) Relations in the plane (0,q). One may write u = ul(?) + u2(p) and analogous formulas for v and w, X and relations (I.19). One has in particular dUl 3u + u du2 u dX 5q 60 du o0 Besides, according to equation (1.20)  being defined by the P cos T dul + cos 0 dvl + sin 0 dwl = 0 (1.21) 3 cos T du2 + cos 0 dv2 + sin 0 dw2 = 0 however: 0 = x + P, T = A 1; and consequently the first equa tion (1.21) is written cos r 1 cos K dui + cos K dvI + sin k dwli + sin psin du sin K dv1 + cos K dw = 0 since the two quantities between brackets are unique functions of the preceding equality causes p cos K dul + cos K dvi + sin X dwl = 0 P sin k dul sin X dv1 + cos K dwI = 0 dv1 p dul = 1 cos 2X dwl sin 2X (1.22) NACA TM 1354 In the same manner one will show that dv2 dw2 p du2 =  cos 2Ii sin 21 (1.23) (b) Relations in the plane (e,t). The calculation is perfectly analogous. The equation (1.16) causes us to introduce the complex variable 0 = 8 + it and the func tions U(.), V(C), W(t), defined with the exception of an imaginary additive constant, the real parts of which in (A) are, respectively, identical to u(e,a), v(0,t), w(e,t). The equation'(I.20) permits one to write 0 ch t dU + cos 0 dV + sin 0 dW = 0 If one puts e + it = e iE5 = one obtains cos T L 2sin sin [ cos dU + cos  2 2 sin dU sin  2 2 dV + sin dW + 2 dV + cos  2 thence one concludes as previously  dU dV dW cos ( sin c The formulas (I.22), (1.23), (I.24) express the relationships of compatibility which we had in mind. dW = 0 (1.24) NACA TM 1354 Remark. We shall utilize frequently the conformal representation for studying the problems relative to the domain (A). If one puts, in particular Z = eit = e ei one sees that (A) becomes in the plane Z the interior area of the circle (CO) with the center 0 14 and the radius 1 (fig. 5). If one puts Z = pe the point Z is the image of a semiinfinite line, issuing from the origin of the space (xl,x2,x3), characterized by the angle 6 and the relationship X1_ 1 + p2 pr 2p The origin of the plane Z corresponds to the axis of the cone (P), the circle (Co) to the cone (r) itself. A problem of conical flow appears in a more intuitive manner in the plane Z than in the plane f. In the plane Z, the formulas (1.24) are written SdU = 2Z dV = 2iZ dW (1.25) Z2 + 1 Z2 1 We shall moreover utilize the plane z defined by S 2Z Z2 + 1 The domain (A) corresponds conformably to the plane z notched by the semiinfinite lines Ax, A'x' (fig. 6), the cone (r) at the edges of the cuts thus determined, and the axis of the cone (F) at the origin 14 No confusion is possible between the point 0, origin of the sys tem of axes xl, x2, x3 and the point 0, here introduced as the origin of the plane Z. NACA TM 1354 of the plane z. The relations of compatibility in the plane z then assume the form 0 dU = z dV iz dW (1.26) 1 z2 1.2.6 Boundary Conditions The Two Main Types of Conical Flows The boundary conditions are obtained by writing that the velocity vector is tangent to the cone obstacle. Let, for instance, x2(t), x3(t) be a parametric representation of the section xI = 0 of the cone; x3x2' x2x'3, xx3', PY2' constitute a system of direction parameters of the normal to the cone obstacle, and the boundary condition reads x' vx3' = (x3x2' x2x3' (1 + u) (1.27) It will be possible to simplify this condition according to the cases. However, the simplification will have to be treated in a dif ferent manner according to the conical flows investigated. As set forth in section 1.1.2, two main types of conical flows may exist. (1) The flow about cones with infinitesimal cone angles, that is, cones where every generatrix forms with the vector U an angle which remains small. Naturally, the cone section may, under these conditions, be of any arbitrary form; since the flow outside of (P) is undisturbed (velocity equivalent to U), on the cone (r) u, v, w are zero. The problem may have to be treated in the plane Z; U(Z), V(Z), W(Z) will have real parts of zero on (CO). The image (C) of the obstacle, in the plane Z, is defined by a relation p = f(e); conse quently, a parametric representation of the section xi = p will be obtained by means of the formulas 2p 2p X2 = 2 cos e X3 sin G 1 + p2 1 + p2 NACA TM 1354 Thus the condition (1.27) becomes w sin 0 p' cos a + p2(p sin e + p' cos 8) + v cos 0 + P' sin e + 2(p cos 0 p' sin 8) = 2(1 + u) (1.28) with 0 taken as parameter, and p' denoting the derivative of p with respect to 0. The investigation of conical flows with infinitesimal cone angles will form the object of chapter II. (2) The flow about flattened cones, that is, cones, the generatrices of which deviate only little from a plane containing U. Let us remember that (section 1.1.2) the tangent plane is to form a small angle with 6; consequently, rigorously speaking, the section of such a cone cannot be a regular closed curve, an ellipse for instance; it must present a lentic ular profile (fig. 7). In chapter III we shall study the flows about such cones. Remark. Actually, we have, therewith, not exhausted all types of conical flows, that is, those for which linearization is legitimate. One may, for instance, obtain flows about cones, the section of which presents the form shown in fire 8; the axis of such a cone has infinitely small inclination toward U. Before beginning the study of these flows we shall, in order to terminate these generalities, introduce a generalization of the flows, the possible utilization of which we shall see in a final chapter. 1.3 Homogeneous Flows 1.3.1 Definition and Properties The conical flows are flows for which the velocity potential is of the form cP = rf(e,X) as we gad seen in section 1.2.5. One may visualize flows for which P = rnf(0,x) NACA TM 1354 We shall call them homogeneous flows of the nth orderl5. The conical flows defined in section 1.2 are, therefore, homogeneous flows of the order I. However, we shall maintain the expression "conical flow" to designate these flows since this term has been used by numerous authors and gives a good picture. The derivatives of the velocity potential with respect to the vari ables xl, x2, x3 all satisfy the equation (I.10). If one then con siders the derivatives of the nth order of the potential of an homogeneous flow of the nth order, one finds that they depend only on X. and a and satisfy the equation (1.14); the analysis made in section 1.2.2 remains entirely valid. One may make the changes in variables (I.15) and (T.17) which lead to the equations (1.16) and (1.18). Thus one has here a method sufficiently general to obtain solutions of the equa tion (I.10) which may prove useful. The simplest flows are the homogeneous flows of the order 0 which do not give rise to any particular condition of compatibility. For the flows of nth order, in contrast, one has to write a certain number of conditions connecting the derivatives of nth order. We shall examine16 as an example the case of homogeneous flows of 2nd order. There are six second derivatives which we shall denote Tij (i and j may assume independently the values 1, 2, 3), Cij designating 4 Outside of (r) we shall put 6xi 6xj ij ij i 1 2 with P.ij being a function of X only, Pij2 of p only (see for mula 1.19). Inside of (r), ij. is the real part .of a function .i.(U). In order to obtain the desired relations, it is sufficient to note that 15The definition for homogeneous flows of the nth order has been given for the first time by L. Beshkine (ref. 11); this author, by the way, calls them conical flows of the nth order. One may also connect this question with the article of Hayes (ref. 12). S6ee appendix 2. NACA TM 1354 ij dxj = dPi and to apply the results of section 1.2.5; thus one may write the fol lowing six relations between the c 1 0 dil 1 d 1 d 1 (i = 1,2,3) 11 cos 2X 12 sin 2X i3 which, besides, are reduced to five as one sees immediately. One will have analogous relations for the functions Cpj2 (it is sufficient to exchange the role of X and of p). Finally, one has for the analytic functions oij() d 1 d 1 d3 ii cos i i2 sin i i3 namely six relations which as before are reduced to five. The written conditions are not only necessary but also sufficient since the func tions Pi necessarily are the components of a gradient. Thus one sees that there is no difficulty in writing the conditions of compatibility for a homogeneous flow of nth order. 1.3.2 Relations Between the Homogeneous Flows of nth and of (nl)th Order We shall establish a theorem which can be useful in certain prob lems and which specifies the relations existing between homogeneous flows of nth and of (n1)th order; we shall examine the case where n = 1. 1.3.2.1. Let us consider inside of the cone (P) a homogeneous flow of the order 0 defined by f = REcZ)] NACA TM 1354 2 We shall first of all seek the components u, v, w of the dis turbance velocity dC = u dxI + v dx2 + wdx = R[O'(Z) dz = R[Zs'(Z) d then 1 + p2 xl = pr 2 2P thus dZ = dp + i d0 Z p x2 = r cos 0 x3 = r sin 8 _ 2 + 1 dx x2 dx2 + x2 dx p2 1L r2 Sx2 dx3 x3 dx2 r2 whence one deduces p2 + 1 1 p2 1 1 v cos P02 + 1 r P2 1 w sin 8 02 + 1 r p2 1 R[Zq'(Zz] + sin e T[ Z'(z] RZo'(Z CS '(Z ~r  p2 + 1 sin 0 P p21 sin 0 P however Z 1 z Z z p2 + 1 P21 _ 2  p cos 8 + i cos e + i NACA 4M 1354 hence the result 1 u xl xl p2 + 1 R Z'(Z) p2 p2 +1 p2 1 p2 1 R [ (Z2 + 1)'(Z) R (2 1)0,(Z)] R 2 1)01(z (1.29) 1.3.2.2. Let us now consider a point O' (xI = E1, x2 = 0, x3 = 0), El being a very small quantity. Let M be a point with the coordi nates (xl,r,e) with respect to O, inside of (F), and with the para meters (p,8) in the plane Z. For the conical flow (homogeneous of Ist order) with the vertex 0', its coordinates are: (xl E r, a) and its parameters in the Zplane: ( p2 1 1 since dxI = 1 = Br p2 1 2p2 dp = x 2 1 dp do = x p p + 1 Let us then consider two identical conical fields but with the apexes 0 and 0', and form their difference. We shall obtain a velocity field which, due to the linear character of the equation (I.10), will satisfy this equation. If U0 = R[(Z) denotes the component u of the field with the vertex component u in the "difference field" 0, one has as u = +RF(Z) R 2 + 1 z P2 X1 ]  P2 + 1 1 RFZF'(Z) p2 1 xl ~~2 (I.30) NACA TM 1354 El being considered as infinitely small. Moreover, according to the relations (1.25), the components v and w are written v = 2 + 1 aE fi (Z2 + 1)F'(Z] w l p2 + i PZ2 1)F'(Z) xi p2 1 2 1.3.2.3. Let us consider the point 0''(O,E2,0), with E2 being a small quantity.' Let M be a point with the coordinates (x1,r,O) with respect to 0, inside of (r), with the parameters (p,8) in the plane Z. For the flow with apex 0'', the coordinates of M are (x1, r E cos 8, 0 + E2 sin as can be easily stated by projecting M in m on the plane x2x3 (fig. 9). But on the other hand 2xd 1 2 dr dp = C2 cos 0 ( (1+p2)2 r de x P dO = 2 sin 0 1 + p2 thus dZ = e p + ip d] = e2 12 2 x i sin e cos 8 + e1i with Z + dZ representing the point M in the conical field with the vertex 0''. Let us then consider two identical conical flows, but with the apexes 0 and 0'', and form their difference. We shall obtain a velocity field which due to the linear character of the equation (I.10) will satisfy this equation. If O = R[G(Z) 28 NACA TM 1354 denotes the component v in the field of the vertex 0, one has a com ponent v in the "difference field" v +n[G(Z) R[(Z + dZ) = R[G'(Z)dZ] Cx2 P2 +_1 2) + i sin 1(P2 = p R G'(Z) cos e ( + p2) + i sin e(p2 l)e S2 P 2+ 1 R ZG'(Z) Z +1 2x1 2 1 2 2x p2 1L Z/j x1 P2 1 2 (1.32) besides, according to equation (I.25), the components u and w are written E2 02 + 1 1 po' 1 (1.33) w =2 p2 + 1 R (Z2 1)G'(Z] x p2 1 2 1.3.2.4. With these three lemmas established, it is easy to demon strate the property we have in mind. Let us call "complex potential" of a homogeneous flow of zero order the function ((Z) (section 1.3.2.1) so that P = R [0(Z)] so that the function of complex variable, the real part of which gives insideof (P) the projection of the disturbance velocity in the direc_ tion i, is the "complex velocity" of a conical field in the direction 1; so that, finally, the velocity field obtained by the difference of two iden tical conical fields, the verties of which are infinitely close and ranged on a line parallel to 1, is the "field derived from a conical flow" in the direction 1; then we may state: NACA TM 1354 Theorem: The field derived from a conical flow in the direction Z is the velocity field of a homogeneous flow of zero order; the complex potential of that flow of zero order is proportional to the complex velocity of the conical field given in the direction i, since the pro portionality factor is real. The proof follows immediately. According to sections 1.1.2 and 1.1.3 one may be satisfied with considering, for definition of a homogeneous flow, the inside of the cone (r); comparison of the for mulas (1.29), (I.30), (I.31), (1.32), (1.33) entails the validity of the above theorem'if I is parallel or orthogonal to U. Hence the general case where I is arbitrary may be deduced immediately; if F(Z), G(Z), H(Z) are the complex velocities in projection on Oxl, Ox2, Ox3, the expression for the component u of the field derived in 4 the direction I(EE 2,3) is u = p2 + 1 R Z F'(Z) + e20'(Z) + 3H'(Z) u i 02 1 C Thus, with E1F(Z) + E2G(Z) + E3H(Z) being the complex velocity in projection on 1, comparison of this formula with the first formula (I.29) completely demonstrates the theorem. 4 Corollary: The field derived in the direction 2 of a conical flow, the complex velocity of which in the direction I is K(Z), is a velocity field of a homogeneous flow dependent only on K(Z) (not on the direction I). The theorem just demonstrated may be extended without difficulty to the homogeneous flows of nth and (nl)th order. A statement of this general theorem would require only specification of a few definitions; however, since we shall not have to utilize it later on, we shall not formulate this statement. NACA TM 1354 CHAPTER II CONICAL FLOWS WITH INFINITESIMAL CONE ANGLES * 2.1 Solution of the Problem 2.1.1 Generalities We shall now treat the first problem set up in section 1.2.6. We shall operate in the plane Z. Let us recall that the image of the cone (r) is the circle (CO) of radius unity centered at the origin, and that the image of the obstacle is a curve (C), defined by its polar equation p(e). We shall denote by (D) the annular domain comprised between (C) and (CO); we shall call (O0) the circle of smallest radius centered at the origin and containing (A) in its interior, and we shall call k the radius of the circle (70). In this entire chapter, k will be considered as the principal infinitesimal. The problem then consists in finding three functions U(Z), V(Z), W(Z) defined inside of (D) except for an additive imaginary constant, so that (1) 3 dU 2Z dV 2iZ dW (1.25) Z2 + 1 Z2 1 (2) the real parts u, v, w, which are uniform become zero on (CO), (3) on (C), one has the relation vp cos e + p' sin 8 + p2(p cos e p' sin e) + r j 2 p sin e p' cos e + p2( sin 0 + p' cos e) = 22( + u) Put in this manner, the problem is obviously very hard to solve in its whole generality; however, an analysis of the permissible approxima tions will simplify it considerably. 2.1.2 Investigation of the Functions U(Z), V(Z), W(Z) 2.1.2.1. An analytical function of Z will be the said func tion (A) if its real part becomes zero on (Co). Let us designate by NACA editor's note: Some minor inconsistencies appear in the number of equations in this chapter and subsequently in chapters III and IV, but attempt was made to change the numbering as given in the original text. NACA TM 1354 (70') the circle with the radius 1/k, centered at the origin, and by (D') the annulus limited by (o0) and (70') (fig. 10). Lemma I. A uniform function (A), defined inside the annulus limited by (70) and (CO) may be continued over the entire domain (D'). This results immediately from Schwartz' principle. Let M and M' be two symmetrical points with respect to (Co), M being inside of (CO); 6ne defines the function (A) at the point (M') as having, respectively, an opposite real and an equal imaginary part compared to the real and the imaginary part of the function given at the point M. Lemma II. A holomorphic function (A) inside of (D') has a Laurent development of the form17 + 2IQnn~izn ip+ Z Kn) 1 Let H(Z) = h + ih' be such a function (A). Let us write its Laurent development in (D') provisorily in the form H(Z) = JnZn + n 0 1 It is an immediate demonstration and yields the formulas defining Jn and Kn Kn = (h + ih')o7eind8 21t 0O 17 We remember that Kn denotes the conjugate imaginary of Kn. NACA TM 1354 (h + ih')70 denoting the value of H on (70); likewise J kn 2 I230 (h + ih') ieinede (h~ih '70 Consequently, according to the lemma I: Kn = Jn moreover S2n H(Z) d Z 2xJo S.2n (h + ih')0 dO  00 2n 0o is purely imaginary, and the lemma II is therewith demonstrated. We shall note that, if H(Z) is limited by M on (70) one has the inequality or (70'), Kn < Mkn (II.] Lemma III. A function (A) with a real and uniform part defined in (D) can be developed inside of (D') in the form B log Z + ip + (f KnZn (II._ with B being real. Actually, the derivative of the function (A) is necessarily uni form. Thus one knows (see for instance ref. 13) that one may consider the given function as the sum of a uniform function H(Z) and a loga rithmic term; since the critical point of the logarithm is arbitrary inside of (O), it is particularly indicated to choose this point at the origin; since the real part of the function is uniform, the coeffi cient of log Z is real. Besides, since log Z has a real part zero h d Co Jo i 2i ,0 co NACA TM 1354 on (Co), H(Z) is itself a function (A). The given function may therefore be continued inside of (D') and the development (11.2) is thus justified. Remark. (7o) form If one chooses as pole of the logarithmic term a point inside of but different from the origin, one obtains a development of the B' log aa  + i8 + 1 1  K'nZn 2.1.2.2. The functions U, three functions (A) with a real be developed in the form (II.2).  e u(z) 2 = A log Z + ia V(Z) = B log Z + ip W(Z) = C log Z iy + ixz V, W of the variable Z are all uniform part and, consequently, can We shall write henceforward (n JZ Znn nZn) J K n Zn  n  L,'n Lnzn) (11.3) A, B, C are real, a, 0, ? are real and also arbitrary; but these developments are not independent since the relations (I.25) must be taken into account. For instance, Z dV/dZ must be divisible by Z2 + 1; otherwise we would have for U logarithmic singularities on the cone (r) which is inadmissible. Now =Bi 1 n(K"n + izZn Zn + K, I zd dZ (+ Ln 1 NACA TM 1354 Hence one deduces the relations B 1= ( P[K2p + R2] 1 S=3 ( 1)P(p + 1)K2p 0  Kp+1 obtained by putting in the preceding equality Z = i and Z = i. Z dW/dZ must be divisible by Z2 1 which gives C = 2p (L2p + L2p) 1 0 => (2p + 1)(L2p+1 + L2p+l) 0 Finally, the qualities (1.25) lead, in addition, connecting the coefficients of the developments (II.3) thus one may write the relations B + 2K2 =i[C 2L2] nKn (n 2)Kn2 = to relationships among themselves; K1 K1 = i + L] i[(n 2)Ln2 + nr] (n >,2) and on the other hand B =(J + J1 K1 = A + 2J2 nKn = (n 1)Jn1 + Likewise, (n >2) (n + 1)Jn+1 NACA TM 1354 2.1.2.3. Approximations for the developments (II.3). Moreover, the hypotheses of linearization must be taken into account which, as we shall see, will permit us to simplify the developments (11.3) consider ably and will lead us in a very simple manner to the solution of the problem posed in section 2.1.1. The qualities (11.6) make V(Z) and W(Z) order. We shall denote by M an upper limit of circle (70). M will be equally an upper limit (70') and hence in the entire domain (D'). seem of the same their modulus on the of their modulus on If one utilizes the inequality (1.1), (11.4) shows that18 B = 0 Mk2) K1 1 = o(Mk2 If one assumes a, 0, 7 zero in what follows, which does not at all impair the generality, one may write the second formula (11.3) in the form V(Z) R(Kn (  and consequently: 2 2 Kn = B log Z  Zn In the annulus limited by (70) and equality is o(Mk2log k) Likewise according to equation (11.5) C = o(Mk2) (Co), the second term of this L1 + i =o(Mk3) W(Z) iT(L1)(. + Z) n = C log Z + RL) 2  z)  n 2 180 denotes Landau's symbol, A = 0 Mk2) signifies that A is Mk2 limited when k tends toward zero. nZn + iT(K1)( + z) NACA TM 1354 In the annulus comprised between (70) and (CO), the second term of this equality is also S(Mk21og k) Furthermore, according to equation (1.6) Kn2 + Ln2 = 0 (kn) (n > 2) Thus W(Z) iV(Z) = (Mk2log k) + 2iK1Z in the annulus (70,C). Finally, according to equation (11.7) A = K1 + o(Mkl3) Jn = n 1 Knl + O n+2 n Q U(Z) = R(Kl)log Z 2K2Z + n + 1 Kn+ +1 2  n Zn 1 Summing up: If one is satisfied with defining V(Z) for O(Mk2log k) and U(Z) except for O(Mk3log k), the corona (o',Co) O(Mk3log k) and W(Z) except one may write in W(Z) = iV(Z) + 2iKiZ V(z) = H(Z) K1Z (11.8) Thus NACA TM 1354 37 with H(Z) = Kn (II.10) 1 and U(Z) = Z dZ 2K2 (II.11) The coefficient K1 may be supposed to be real, and the integra tion occurring in equation (II.11) must be made in such a manner that R[U(Z)] will be an infinitely small quantity of the third order at least on II = 1. 2.1.2.4 Remarks. (1) The formula (1.8) which is the most important may be estab lished immediately from the second formula (1.25). However, the method followed in the text, even though a little lengthy, seems to us more S natural; also, it shows more clearly the developments of the func tions U, V, W. (2) Strictly speaking, the hypotheses set forth in the course of this study must be verified by the solutions found in each particular case. We shall, however, omit this verification which in the usual cases is automatically satisfactory. (3) The results obtained by the preceding analysis and condensed in the formulas (II.8), (II.9), (II.11) are in all strictness valid only in the annulus (o0'CO), but not in the domain (D). However, it is very easy to extend, by analytical continuation, the definition of H to (D). Let us now first suppose that (C) contains 0 in its interior; since one may write V(Z) in the form V(Z) = H(Z) Z KnZ + B log Z 1 one sees that, since V(Z) is defined by hypothesis in (D), and one can extend KnZn and B log Z inside of (70) up to (C), H(Z) 1 NACA TM 1354 may itself be defined without difficulty inside of (D). The case where (C) does not contain the origin offers no difficulty; it is then suffi cient to utilize the development given at the end of section 2.1.2.1. As to the order of the terms neglected when one writes the equal ity (IT.9) in the domain (D), they are found to be O(Mk2log k) in (D) in the case where there exists inside of (C) a circle of the radius Xk (X and 1/k may be considered as 0(1)). Besides, if that is not the case, one may justify the validity of the results of the formulas (11.8), (II.9), (11.10), (II.11) by making a conformal representation of the domain (D) on an annulus; the radius of the image circle of (Co) may be assumed equal to unity; the image circle of (C) has a radius infinitely small of first order with respect to k and the study may be carried out in the new plane of complex variable thus introduced, without essential complication. 2.1.3 Reduction of the Problem to a Hilbert Problem If one puts, according to the formula (II.8) V = v + iv' with v' denoting the imaginary part of V, one may write on (C) the relation w = v' Since one may, of course, with the accepted approximations, neglect u compared to 1 in the second term of the formula (1.28), one sees that this boundary condition (1.28) affects now only one single analytical function, the function V(Z); this is a first fundamental consequence of the preceding study. Formula (II.9) shows that this condition con sists in posing a linear relation between the real and the imaginary part of H(Z) on the obstacle. Now according to equation (II.10) the function H(Z) is a holomorphic function outside of (C), regular at infinity; the problem stated which initially referred to an annular area (D) is thus reduced to a Hilbert problem for the function H defined in a simply connected region; exactly speaking, one has to solve an exterior Hilbert problem. This is the second fundamental consequence of the results of section 2.1.2. Since we attempt to calculate V(Z) and W(Z) not further than within O(Mk2log k), and U(Z) within 0(Mk5log k), the relation (1.28) NACA TM 1354 which is written R (v iw)2Z2d0 i dz(l p2 may be simplified and reduced to S dZ(v iw) 2P d [ ] 2 On (C), and therefore KiZ is, according to equation (II.1), of the order of Mk2, H = V = v + iv' = v iw consequently, H satisfies, on (C), the Hilbert condition R iH(Z) dZ] 2p d6 (11.12) 2.1.4 Solution of the Hilbert Problem A function H(Z), holomorphic outside (C), regular and zero at infinity, satisfying on (C) the relation (II.12) must befound. Let a1 z = Z + a+ + S Z (11.13) be the conformal canonical representation of the outside of (C) on the outside of a circle (7) centered at the origin of the plane z; the adjective canonical simply signifies that z and Z are equivalent at infinity. On (7) we shall put z = reil S2p2 d . de NACA TM 1354 r being constant and well determined. Let us put F(Z) = i log z (II.14) One has on (C) or on (7) F'(Z) dZ = i dz = dP = f(e) dO (II.15) z with f being real; consequently d d F'(Z) d F'(Z) P ,z =~ 4i and therefore equation (II.12) is written H(Z) p2 S2 (11.16) l F'(Z) p dI H(Z)/F'(Z) is a holomorphic function outside of (C) and regular at infinity. Following a classical procedure, we thus have reduced the Hilbert problem to an exterior problem of Dirichlet. Let G(Z) be the holomorphip function outside of (C), real at infinity; its real part assumes on (C) the values G(Z) is 0 dQP determined in a unique manner. According to equation (II.12) H(Z) = iG(Z)F'(Z) + iEF'(Z) (11.17) with c being a real constant. However, we have seen (section 2.1.2.3) that the coefficient of 1/Z in the development of H(Z) around the point at infinity (coeffi cient KI) was real; now, around the point at infinity a iF(Z) = + +. Z 2 NACA IT 1354 41 In order to have the development of the second term of the formula (11.17) admit a real coefficient of 1/Z, E must be zero since G(Z) is real at infinity. Thus the desired solution is H(Z) = iG(Z)F'(Z) (II.18) With the function H(Z) thus determined, the formulas (11.8), (11.9), (II.11) permit calculation of the complex velocities U(Z), V(Z), W(Z) within the scope of the accepted approximations. Thus the problem posed in section 2.1.1 is solved. Remarks. (1) Uniqueness of the solution. The preceding reasoning shows the solution of the Hilbert problem satisfying the conditions (II.16) to be unique. This result will be valid for our problem if one shows that every function satisfying the condition (II.16) is a solution of the initially posed problem (condition (II.14)) which is immediate since it suffices to repeat the calculation. (2) Calculation of the coefficient K1. According to what has been said above, the coefficient K1 is equal to the (real) value assumed by SG(Z) at infinity. In order to find G(Z), we may solve the Dirichlet problem in the plane z; according to a classic result of the study of harmonic functions, K1 is equal to the mean value of 2p2 O on the circle (7). Hence K1 = 2f 2p d d9P = if p2d = 2S 27 0 d dp PJ() Tr wherein S represents the area inside the contour (C). 2.2 Applications 2.2.1 General Remark Let us consider a cone of the apex 0 in the space (Oxl,x2,x3), the image of which in the plane Z is the curve (C), defined by its polar equation p(e). According to the definition of p (see the remark of section 1.2.5) the sections of this cone made by planes par allel to Ox2x3 are homothetic to the curve NACA TM 1354 x2 2p cos e x 2p sin 8 (II.19) 1+ p2 1 + p In the case of the linear approximations, with grad u, grad v, grad w being infinitely small (it would even be sufficient that they should be limited), one sees that one may, within the scope of the approximations of section 2.1, simplify the formulas (11.19) without inconvenience and write them x2 = 2p cos 6 x3 = 2p sin 0 hence the result, essential for the applications. The curve (C) in the plane Z is homothetic to the sections of the cone obstacle made by planes normal to the nondisturbed velocity. Let us likewise consider a cone with variable but small incidences so that the flow about the cone should always be a flow in accordance with the hypotheses of this chapter. One sees that if the orientation of the cone varies with respect to the wind, the curve (C) in the plane Z undergoes a translation. 2.2.2 Study of a Cone of Variable Incidence This last remark allows us to foresee that when a thorough investi gation of a cone has been made for a certain orientation with respect to the velocity it will not be necessary to repeat all the work for any other orientation. This we shall specify after having demonstrated the following lemma. 2.2.2.1 Lemma. One may write on (C) that 2P2 de 2 R Z dZ (II.20) 0P d r;, ~d (11.20) P dT p dz Actually, let us put Z = p cos 0 + ip sin e = X + iY X and Y may be considered as functions of :P. 42 NACA TM 1354 Hence one deduces that Y tan e = X de cos2 Y'cpX X'rpY X2 x2 and consequently 2 p2 de = 2 Y' X X'rY) = 2 R P a;P 134 p/  i dZ 2 R dZ d;' R dz which establishes the formula (11.20). 2.2.2.2. Let us now consider two contours (CO) and (C1) defined in the plane Z by two functions Z(O)(') and Z(1)(c) such that Z(O) = Z(1) + a, a being a complex constant determining the change in orientation. In the development (II.13) which gives the conformal repre sentation, only the coefficient a0 varies when one passes from the contour (CO) to the contour (Cl). Consequently ,z (1) dz dZ(o) dz and the Dirichlet condition determining the function GC"'(z) is written in the plane z RE()z) =2 R z(l) z] L )z + Raz dz_ 0 dz] dz (we have omitted superscripts for the quantities which retain the same value, affected by the index 0 or 1). Consequently G(1)(z) = G(0)(z) + [g(z) since g(z) is a regular function and real at infinity, holomorphic out side of (7), the real part of which on (7) assumes the NACA TM 1354 values R(az dZ/dz), g(z) is then very easily determined. One has exactly g(z) = z dZ + ar2 \dz / z Thence for the function H(1)(z), (since F'(Z) = i/z dz/dZ) H(1)(z) = H(0)(z) + a dz + 2 0 \ dZ( 2 dZ (11.21) The formula (11.21) gives immediately the solution of the problem of change in orientation with respect to the nondisturbed flow. 2.2.3 Cone of Revolution We shall study first of all the case of the cone of zero incidence. One may then do without the preceding analysis and obtain the solution directly; that is what we shall do here. The curve (C) is a circle of the radius p = cte = r; the relation (1.28) is written 2rO v cos e + w sin 8 = 2ro P(1 + r02 On the other hand, for reasons of symmetry v sin 0 w cos e = O Hence one deduces immediately the values of v 2r0 cos 0 S= b0 1 r02 and w on (C) 2r0 sin 0 1 r02) NACA TM 1354 whence v(z) = 2ro 2 Z 3(1 ) r W(Z) = i 2ro 2 0 1 ro + ) (11.22) Finally the relations (1.25) permit the calculation of U  dU 2r02 2Z 11 + Z2 _ P(l r) Z2 +1 Z2 4 rO2 1 S1 ro4 z 2 U(Z) 4 r0 log Z 2 1 r4 (11.23) We shall now study, returning to the method of section 2.1, the case of a cone of revolution with incidence. The formula (II.13) is written z = Z a a being a constant which may be supposed to be real. Consequently F'(Z) = i On the other hand, an immediate calculation shows that do r(r + a cos T) dP p2 whence NACA TM 1354 and consequently whence 2p2 dO 2 (2 + r cos rP) P d'P (\ G(Z) = (r2 + ar2) P\ Z a According to equation (11.18) H(Z) = 2 + ar2 ) 1 2 r2 Z \ Z aZ a p (z a)2 the calculation is easily accomplished; one finds V(z) 2r2 Z 1 (Z a)2 U(Z) = 4 og(Z a) 02 3_a a2 Z a (Z a)2 K2 = +a r2 In particular, one finds, if a = 0, by means of the approximate formulas (11.24) and (11.25), the same result as by the formulas (II.22) and (II.23) under the condition of neglecting in these formulas the term in r0 of the denominator. In order to give to these formulas a directly applicable form it suffices to again connect the quantities a, r with the geometrical data; for this purpose, one must use the formula defining p (p. 42). since + 4aZ (11.25) NACA 3T 1354 Figure 11 represents the cone section made by the aerodynamic plane of symmetry; a is the semiangle at the apex, 7 denotes the angle of the cone axis with the nondisturbed velocity. One has immediately 2r = pa 2a = py Finally, we shall utilize for the calculation of Cp the for mula (I.11) since the velocity component u is infinitely small com pared to the components v and w. This formula is here written Cp = 2R[U(Z) V(Z) 2 (11.26) According to equations (II.24) and (11.25) one has C = 2a21og ~2 2 + 4ay cos 0 + 272cos 20 (II.27) jp Pa The case of the cone of revolution of zero incidence is obtained by making 7 = 0. One finds then again a known result. The for mula (II.27) had already been given by Busemann (see ref. 9) without demonstration. 2.2.4 Elliptic Cone We assume first of all the simplest hypotheses where the planes Oxlx2, Oxlx3 are symmetry planes of the flow (U is in the direction of the cone axis), with the cone flattened out on Oxlx2. The formula (II.13) may be written in the form Z = z + a2 z or p cos 6 + ip sin = r + a2 cos 9 + ir a)sin 9 r r \ 48 Hence one deduces successively tan B = r2 a2 tan P r2 + a2 Scos2 r2 a2 cos2m r2 + a2 NACA TM 1354 r2 a 1 r2)2 r 2 2 dO = 2/2 a 0 d' P r2 The Dirichlet problem, which permits calculation of G(Z), is readily formulated; since G(Z) has a constant real part on the con tour (C), G(Z) is constant: z2 F'(Z) = i 1 z z 2 whence H(z) = r2 H(Z) 2 (r _ a2 a2 z aA 1 r2Z2 4a2 We note besides that and K2 = 0. NACA TM 1354 One calculates V(Z) by the formula (II.9) (14a V(Z) = 2(r2 1 Z) \ r2/ ^" j2_ and U(Z) by the formula (II.11) which may also be written U(Z) = H K G 2K2Z 0 1 z ' whence U(z) = i2 r p2\( U(Z) 2 p2 lo z z2 a2 Z + Z2 4a2 2 If one makes a = O, one will find again the expressions already obtained for U(Z) and V(Z) in the case of a cone of revolution of zero incidence (formulas (II.24) and (II.25) in which one makes a = 0). We shall denote by e and by Tj elliptic cone (see fig. 12). One has the principal angles of the E3 = 2(r + a2 T = 2(r a) whence r = (E + r1) 4 (11.28) (11.29) or (11.30)  l r 2 a2 = 16 (C2 2) NACA TM 1354 The pressure distribution on the cone circumference is easily cal culated. It is sufficient to apply the formula (II.26); besides IV(Z)2 = E2112 T2cos2p + E2sin2cp R[U(Z] = log C+ T 2 2sin2(2 r2cos2C + E2sin2rg hence the final formula  1 + The case where the velocity is may be treated equally by utilizing one must put HO(z) = (2 a 1 <  One then obtains H(l)(z) = 2(r2 (r r z2 a2 not in the direction of the axis the formula (11.21). In this formula dZ= 1 a2 a2 dz z2 z2 z2 a2 zr2 z2 z2 z2 a2 2 kr2 E z + r2 a2] P(z2 a2) k 2 hence, remarking that a2 Z = z +  + a z and (11.31) NACA TM 1354 H(Z) = 2 2 1 \ r T(Z )2 4a2 (ax2 a2(z a( (Z a)2 4a2 2a2 (Z a)2 a2 V(Z) = H(Z) jr2 Z 2] On the other hand, we shall calculate U by utilizing the vari able z and the formula (II.20). The coefficient K2 is equal to K2 ( r2 a + r2 a 2a and U(z) is then given by the formula a4 og 2a2 + az _ log z z r2/ z2 a2 + a2 2 (22 + a) z(z2 a2) One will note that, if mula (11.30), and that, for except for the notations. one puts a = 0, one finds again the for a = 0, one finds again the formula (11.25), Thus one can, without any difficulty other than the lengthy writing expenditure, calculate the pressure distribution coefficient on the elliptic cone of any arbitrary orientation with respect to the wind. U(z) = 4 (r2 P2 K2 +4 Z p (11.32) 52 NACA TM 1354 2.2.5 Calculation of the Total Forces We have already seen in section 1.2.6 that the normal to the conical obstacle directed toward the outside has as direction parameters i(x3x2' x2x'), x3', x2' Let n be the unit vector coincidental with this normal, s be the area of the section with the abscissa xl, L the length of this section; one may make correspond to the resultant of the forces acting on a section (x4, xl + dxl) a dimensionlesss) vector Cz = C pn ds (11.33) situated in the plane x2x3, and a dimensionless number C 1 Cp(nU)ds (11.34) the vector Cz characterizes the lift, the number C, the drag. The integrals appearing in the formulas (33) and (34) are taken along the section. Naturally Cz and Cx are independent of this section. One may also replace C by a complex number Cz, the real and iaginary parts of which are equal to the components of the vec tor C on Ox2 and Ox3. For calculating equations (11.33) and (1.34 one may utilize the section xl = 8. If we assume I to be the length of the contour (C) in the plane Z, we may write, taking into account the habitual approximations Cz Cp dZ (11.35) and Cx [i CpZ dZ (I.36) 51 JC J NACA TM 1354 with the integrals appearing in equations (11.35) and (11.36) taken in the plane Z. These integrals present a certain analogy to the Blasius integrals (ref. 13); Cp is given by the formula (II.26); unfortunately, it is not possible to give simple formulas for the total forces since the integrals (11.35) and (11.36) make use of all coefficients of the conformal representation19. We shall apply circular cone; Cp the formulas (11.35) and (11.36) to the case of the is given by equation (11.27) dZ = i fB eide 2 Z dZ = i 24 de 4 2 = xSp One obtains C, = 2ay In the case of the elliptic cone of zero incidence, Cz ously zero  i a2 eiP Z = rei+P + e r Cx = 2( r2 0P Cx= 2a31og 2 3 My2 OM (11.37) is obvi ireicP a2 iPj dZ = i e e doP r  21 Cp dCP Z^O p being given by formula (II.31). Now ScrT dP = 2(Tl2cos2% + e2sin 2 ) + E dt Do T2 + E2t2 19See appendix No. 7. whence with Cp 54 NACA TM 1354 As one can see immediately by putting t = tan CP the calculation of this last integral is immediate. Thus one obtains c = 2 2l g (E + n) 2 (11.38) with 2 being the length of the ellipse with the semiaxes C, . 2 2 2.2.6 Approximate Formula for the Calculation of Cx Let us consider the function U(z); according to formula (II.11) and the remark 2 of section 2.1.4 one may say that the principal term for U(z) is U(z) = 4 Si log z rp2 Consequently, in first approximation 8s C S log r with S being the area inside of the contour (C), and r the radius of the circle (7) on which one makes the conformal canonical repre sentation of (C). If one now calculates Cx, taking into account this approximate formula, one has, according to equation (11.36) C 1S log rR i dZ whence C, =+ 32S2 log_ (II.39) itp3" r NACA TM 1354 We shall state: In every first approximation the value of the drag coefficient C, is given by the formula (II.39). 2.2.7 Case Where the Cone PresenLs an Exterior Generatrix If the contour (C) shows an exterior angular point, the various functions introduced in the course of the study (first paragraph of this chapter) present certain singularities. These singularities we shall specify. Let ZO be the designation angular point of (C), and B6 the angle of the two semitangents to (C) at the point Z0(O < 6 < 1) (see fig. 13); if z0 is the image of the point Z0 in the plane z, one may write, according to a wellknown result, in the neighborhood of zO (LdZ\ K= z zo)k \dz10 with K being a complex constant and k = 1 8; consequently k F'(Z)0 = K1z 0 = 2 (Z ZO 1+k with K1 and K2 being complex constants. F'(Z) thus becomes infinite at the point Z = Z0. In contrast, the function G(z) has, according to definition, a real part which assumes on the circle (7) the values RzZ ] This real part thus remains finite on the circle (7) (and it satisfies there a condition of Holder). According to a known theorem, Sits imaginary part likewise remains continuous on (7) (and likewise Satisfies a condition of Holder). Consequently, one sees, if one refers Sto formula (11.18) that 56 NACA TM 1354 k H(Z) = K3(Z ZO) l+k in the neighborhood of ZO; likewise, U, V, W will, in the proximity of this point, be of the order k with respect to 1 1 + k Z Z Thus the analysis made in section 2.1 is no longer applicable to this case. However, the formulas (11.35) and (II.36) show that if the pressure coefficient assumes very high values in the neighborhood of Z = ZO, the total energy remains finite. According to what we have indicated in section 1.1.3 we consider the solution still valid, with the understanding that the values of Cp in the surroundings of Z = ZO are not reliable. 2.2.8 Delta (A) Wing of Small Apex Angle at an Infinitely Small Incidence If one puts in the formulas r2 = a2, at the end of section 2.2.4, one obtains the pressure distribution on a delta wing with small apex angle. Let us recall that a delta wing is an infinitely small angle. Its angle, according to definition, is the halfangle w at the vertex (compare fig. 14). Thus one has op = 4a The formulas (11.31) and (11.32) are applicable to a delta wing of small angle placed at an incidence also rather small. Let us moreover assume that this opening is infinitely small with respect to the incidence. Under these conditions, the formulas yielding U(Z) and V(Z) are written V(Z) 2 a a Z 42 42 P2 2 4z2 4a2 2 2 U(Z) 4 a 4a + 8a (a a)Z (11.40) 82 2 JZ2 _4a2 02 NACA TM 1354 Actually one is justified in omitting the secondorder terms with respect to a. For calculating Cp it suffices to apply the for mula (11.8); the second term of the second formula (II.40) may be neglected. With the incidence 7, the delta wing being parallel to 70 = 2ia Finally, one may put along the A Z = 2a cos q = R cos (P 2 One then finds c = 2'uy C sin S sin Ox2, one has We remark further that p is related to the angle 213 = mP cos i I of figure 14 W = n cos 2 One may state: the pressure coefficient on a delta wing of infi nitely small opening angle is independent of the Mach number of the flow. One has Cp = 2 if 2i S_ 2 tu uI t if one applies formula (11.35), one finds Cz = inay This coefficient Cz has not the same significance as the one utilized in the theory of the lifting wing. Actually, it is, according (11.41) 58 NACA TM 1354 to the very manner in which it was obtained, relative to the total area of the A (pressure side and suction side); if one takes only one of these areas into account, one must write (neglecting the factor i) Cz = 2nwu This formula has been found by other methods by R. T. Jones (ref. 14). We shall find it again in chapter III, section 5.1.2.4, when studying the general problem of the delta wing which is here only touched on incidentally and for the particular case of a A with infinitely small opening angle. 2.2.9 Study of a Cone With Semicircular Section As the last application, we shall treat the case of a cone with . semicircular section, with the velocity U being directed along the intersection of the symmetry plane and of the face plane of the cone20 (fig. 15). The contour (C) in the plane Z then is a semicircle, centered at the origin, of the radius a (fig. 16). One obtains very easily the conformal canonical representation of the exterior of this contour, on the outside of a circle (7) of the radius r, centered at the origin of the plane z, by means of a par ticular KarmanTrefftz transformation (ref. 13, p. 128) which is written i 2 S6 Z a z re (11.42) Z + a 5n 1i z re a and r are connected by the relationship 4a = 3r41 In order to obtain the correspondence between the circle (7) and the contour (C), one must distinguish two cases. Let us put z = re 20Such a cone formed the front of supersonic models planned by German engineers. NACA TM 1354 (1) < 6 < the corresponding point of of the circle. Let us put under these conditions Z = aeir and we shall find according to formula (II.42): (C) is on the arc tan  2 (11.43) (2) 7 < p < n, the 6 0 ment AA'; let us put under corresponding point of these conditions (C) is on the seg Z = a cos . The formula (11.42) shows that r4 tan  2 P 5t tsin( + 12 sin(+ j (II.44) The two last formulas define completely the desired conformal representation. Figures (17) and (18) give the variations of and x as functions of P. We shall have to utilize equally the value of dz/dZ. The simplest method for obtaining this value consists in logarithmic differentiation of the two terms of formula (II.42). One thus obtains the result dz z2 + rz r2 dZ 2 2 Z r (I1.45) NACA TM 1354 If one has  < ( < 6 6 z = rei Z = aei whence dz r2 1 + 2 sin q ei('A) 8 1 + 2 sin 'P ei((P) dZ 2a2 sin 27 sin * 71r 117 If 'p is comprised between  and , one puts b b Z = a cos X. Thus one obtains i(E A dz 16 1 + 2 sin r e2 ) dZ 27 sin2X The function G(Z) has as its real part R zZ ]d that is [ d~dz 2 ar sin 4 8 1 + 2 sin T 0 if <'JP < 6 6 if 6 < < 1 b o The analytic function a2 z dZ Z dz has a real part which, on (b), assumes these same values. This func tion is regular at infinity, holomorphic outside of (7), but with a pole z = i, with the corresponding residue being equal to ia2 pole z = ir, with the corresponding residue being equal to ia (11.46) z = re , (1.47) (II.48) NACA HM 1354 61 Let us then consider the function a2/2 dZ 1 z ir\ \Z dz 2 z + ir) This function is holomorphic outside of (7). It is regular at infinity; its value at infinity is equal to a2/2. On (7), these real and imaginary parts satisfy Holder conditions. This function is there fore identical with the desired function G(z). Hence one deduces according to equation (II.18) and ac t: H(Z) ( r\a2 dz Z dz 2 z + ir/ z dZ cording to equation (II.19) V(Z) = a2l Z 1 \Z 2 2z Finally, the calculation of U(Z) of formula (II.29) G dz = a2log Z a z ir dz 2 jz + ir z and ZH a2/ 1 Z dz z ir ZH2 z dZ z + 2 z dZ z + ir Sa2 a2 z ir dz Z 2z z + ir dZ z ir dz\ z + ir dZj may be carried out with the aid = a2 (log Z + .1 log z) \ z + ir 2 / S= a2 z ir Z dz\ 2) 2 z + ir z dZ/ whence U(Z) = a2 ir Z dz 1 + + 2 log Z + log \z + ir z dZ g2 z + ir The calculation of the coefficients K2 offers no difficulty soever; however, as one had already opportunity to note, the term does not occur in the calculation of the pressures along the cone. n i; ii i; iii ,i II tl; ,ii ,, ,; what K2 62 NACA TM 1354 This pressure distribution along the cone calculated with the aid of equation (II.26) is represented in figure 19. 2.3 Numerical Calculation of Conical Flows With Infinitesimal Cone Angles 2.3.1 General Remarks In the preceding paragraph, we have studied a certain number of particularly simple cases. However, if the cone (C) is arbitrary, it will be necessary to carry out various operations leading to the solu tion by purely numerical procedures. Let us analyze the various operations necessary for the calculation: (1) The conformal canonical representation of the exterior of (C) on the outside of the circle (?) must be made; this calculation per mits, in particular, determination of the radius r of (7), corre spondence of the points of (C) and of (7), and calculation of the expression dZ on the contour (7). (2) The function G(z), holomorphic outside of (7), regular and real at infinity must be determined, the real part on (7) of which is known; we shall designate it by g('). In fact, it suffices to know, on (7), only the imaginary part of G(z), for instance g'('P); g'(P) is the conjugate function of g(T) and is given by the formula g'() =_ 1 g( ')cot P dc' 2n Jo 2 This formula is called "Poisson's integral." (3) With these two operations accomplished, the values of H(z) on the circle (7) (formula (11.18)) are known which provides the values of v and w on the cone; u is obtained by the formula (11.29). The only new calculation to be made is that of the expression: [B f dz I drP the constant of integration being determined so that u should have a mean value zero on (M). MACA 9M 1354 All these operations always amount to the following numerical problems: (a) With a function given, to calculate its conjugate function (Poisson integral) (b) With a function prescribed, to calculate the derivative of the conjugated function (c) With a function prescribed, to calculate its derivative21. We shall justify this result in the following paragraph by showing that the operation (1) may be performed by applying the calculations (a), (b), (c). We shall then indicate a general method, relatively simple and accurate, which permits solution of these problems. We shall ter minate this chapter by giving an application. 2.3.2 Conformal Canonical Representation of a Contour (C) on a Circle (y) The numerical problem of determination of the conformal canonical representation of a contour (C) on a circle (y) has been solved for the first time by Theodorsen22. We shall briefly summarize the principle of this method, simplifying, however, the initial expos of that author. Let us suppose, first of all, that the contour (C) is neighboring on a circle of the radius a, centered at the origin (fig. O0); in a more accurate manner, putting on (C) Z = ae+ie (11.49) with being a function of e, 0 = 4(e), we shall suppose that *(e) and d are functions which assume small values. We shall then call de 21If the conformal representation of the exterior of (C) on the outside of (7) is known in explicit form, it will naturally be suffi cient to apply operation (a). 22Compare references 15 and 16. One may achieve this conformal representation also by the elegant method of electrical analogies (ref. 17); the time expenditure required by the experimental method and by the purely numerical methods here described as well as the accuracy of these pro cedures are of the same order of magnitude. NACA TM 1354 (C) "quasicircular." Let Q be the angular abscissa of the point of (7) which corresponds to the point of (C), the polar angle of which is 0; we put e = P + c((p) cP = 0 E(0) e(e) and E(cp) representing the same function but expressed as a function of 0 or as a function of qP; we shall put likewise (CP) = J1(e) The desired conformal transformation may be written Z zeh(z) with h(z) being a holomorphic function outside of (7), regular and zero at infinity. The equality (II.50) becomes, if one writes it on the circle (7), ae())+i T +(' = reiqeh(z) whence h(z) = W(~() + iE(cp) + log r r (11.51) Finding the conformal representation of (C) on (T) amounts to cal culating the functions (cp) and T(p). First of all, one knows (equa tion of (C)) that (11.52) (p) = + C(r4) ) On the other hand, according to equation (II.51), e((P) gate function of (cp), and consequently is the conju )=(<'P)cot(' 2 ( N(c') ot M P' k 2 (11.50) e(') 1 ' 2x p n 2v (II.53) NACA TM 1354 the integral being taken at its principal value. There is no constant to add to the second term of equation (11.53), for i(P) has a mean value zero since h(z) is zero at infinity. For the same reason, if *0 denotes the mean value of T('o) in an interval of the amplitude 21 r = ae'0 (11.54) an equality which will permit calculation of r if T(P) is known. In order to calculate T(P) and T(P), one disposes therefore of the relations (11.52) and (11.53); one can solve this system by a procedure of successive approximations. We shall put first o(e) = T0() = o According to equation (11.52) 4(e) = on(0) and according to equation (11.53) l(e) = 1 '(S')cot d' e de' 2 Thence a first approximation for 7' 1 = a l(e) S=' 1 + 1 ) From it one deduces, according to equation (II.52), a first approxima tion for Tl(c') 11 1 + whence a second approximation for the function E NACA TM 1354 r2 1 02n 1 1cot 1' 2p1 1 = j*2y W (c1)cot %i l ,P 2a 2 2 T 2(e) = c2 [e e1()] whence 2 = a E(e) e = 2 + 2n The procedure can be followed indefinitely. The convergence of the successive approximations forms the subject of a memorandum by S. E. Warschawski (ref. 18). We refer the reader who wants to go more deeply into that question to this meritorious report. From the practical point of view one may say that the convergence is very rapid; two approximations suffice very amply in the majority of cases; the different changes in variables which encumber the preceding expose are very easily made by graphic method. Thus one sees that the numerical work essentially consists in calculating twice the inte gral (II.53). This calculation is precisely the object of the prob lem (a) stated at the end of section 2.3.1. If the contour (C) is not "quasicircular," one may make, first of all, a conformal representation which transforms it into the "quasi circular" contour (C'); one will then apply the preceding analysis to the contour (C'). For certain cases it will be quicker to use a direct method. Let us assume, for instance, that (C) is a contour flattened on the axis of the X (compare fig. 21) and for simplification that X'OX is permissible as the axis of symmetry. Let us suppose that X varies along (C) from a to +a while IY remains bounded by ma (with m being, for instance, of the order of 1/10); it will then be indicated to operate as follows: We put along (y). Z = [f(rP) + ig( SNACA TM 1354 One has or also X('C) = f cos CP g sin r Y(P) = f sin p + g cos r f = X cos r + Y sin m g = Y cos '" X sin cP f(P) is an even function of r,, g(C') is an odd function f(0) = +f(n) = a g(O) = g(n) = 0 The functions X(T) and Y(0) have to be found. Let us take as starting point XO(0) = a cos P an approximation which would be definitive if (C) were an ellipse. On the contour (C) one reads the corresponding value YO('P), and by means of the second formula (II.56) one obtains a first approximation gl(+) = YO(cp)cos rp Xo(p)sin (p fl(q) will be given by a Poisson integral 2xi 0 gl(r)cot ' " dT' + X1 2 with X1 being a constant, such as fl(O) = a. Owing to the formulas (II.55), one has a first approximation Xl('), Y1(T) for the functions X(), Y('P). One proceeds in the same manner, reading off on (C) the functions Y1(mP) corresponding to Xl('), then (11.55) (I1.56) NACA I 1354 calculating g2 1) = Yl(c)cos (p Xl(p)sin C and n') =f g2( p')cot P' dTP' + X2 2 When one approximation tions; then has obtained a pair fn(P), gn(') providing a sufficient Xn(), Yn(p) of X(C'), Y('), one stops the calcula r = X In practice23 it suffices to take of very slight adaptations) will apply being flattened on OX, will no longer axis. n = 2; the same method (averaging to the case where (C), although admit of OX as the symmetry Finally, for a complete solution of the problem (1) posed at the beginning of the preceding paragraph, only dZ/dz remains to be calcu lated, which will obviously be possible with the aid of the problems (b) or (c). 2.3.3 Calculation of the Trigonometric Operators24 The method we shall summarize permits calculation of the linear operators A, transforming a function P(e) into a function Q(e) 23The principle of this method is the one we applied for the study of profiles in an incompressible fluid. But in the case of the profiles a few complications (which can, however, easily be eliminated) arise due to the fact of the "tip." 24We gave the principle of this method for the first time in March 1945 (ref. 19). Compare also reference 20. In continuation of this report, M. Watson provided a demonstration of the formulas which we obtained by a different method (ref. 21). We also point out a "War time Report" of Irven Naiman, of September 1945, proposing this same method of calculation for the Poisson integral (ref. 22). etc. NACA TM 1354 69 Q(e) = A[P(e) and reentering one or the other of the following categories: First category: The operator possesses the following properties A(cos me) = am sin me A(sin me) = a cos me (II.57) A(1) = 0 with am being a nonzero constant, m any arbitrary integral different from zero. Second category: A possesses the properties A(cos me) = bm cos me SA(sin me) = bm sin me A(1) = bO with bm being a nonzero constant, m any arbitrary integral. We shall call these operators trigonometricc operators." The operations which form the subject of the problems (a), (b), (c) are, precisely, particular cases of trigonometricc operators." With the function P(e) known, one now has to calculate the func tion Q(e); the functions P(e) and Q(e) are assumed as periodic, of the period 2v. P(e) and Q(8) are determined approximately by knowl edge of their values for 2n particular values of 8, uniformly dis tributed in the interval 0, 2n. One knows that the unknown 2n values of Q are linear functions of the known 2n values of P. The entire problem consists in calculating the coefficients of these linear equa tions. We shall do this, examining two possible modes of calculation. NACA IM 1354 2.3.3.1 First mode of calculation. After having divided the circle into 2n equal parts, we shall put f= f (1) Operators of the first category. Obvious considerations of parity show that the Qi are expressed as functions of the Pj by equations of the form n1 i = KpPip ip) 1 (11.58) We shall apply the equations (11.58) relations (II.57), that is, carry into the P(8) = cos me P(O) = sin me We thus obtain 4n equation Q(e) = am sin me Q(e) = am cos me equations which are all reduced to the unique nl SKp sin p m a 1 (I1.59) This reduction is the explanation for the success of the method. We have to determine (n 1) unknown Kp. For this purpose, we shall write the equation (II.59), for the values of p varying from 1 to n 1. The system remains to be solved. If one multiplies the first 2!, scn b2 i)th equation by sin E, the second by sin L, the (n 1 by n n sin(n 1)n, and if one adds term by term, one obtains a linear rela tion between the Kp, with the following coefficients of Kp NACA IM 1354 71 n1 n1 Wpn 1 (nIP Jfn (p + [Jn sin m sin m  Co cos m n n 2 5 n 1 nn m=l m=l 1 Sn P Cn[ P( + )_n' = 2 E D n In]J with n1 ( sin x Cn(x) = cos mx = cos x 2 x m=O sin Thus the coefficient of Kp is zero if p / n, and equal to if 2 p= U Thence the desired value of Kp nl Kp = a sin m (pI.60) m=l Let us apply this result to the calculation of the Poisson integral. This integral defines an operator Q = A(P) of the first category for which am = 1. Consequently, the formula (II.60) is written n1 Kp = : sin m = Sn mn n n ( 1 if one puts n1 sin nx n (n 1)x s 2 Sn(x) = V sin mx = sin 2 l)x 1 sin 2 1 2 NACA TM 1354 Thus K = 0 if p even (II1.61) 1 cot pA if p odd KP n 2n (2) Operators of the second category. The considerations of parity permit one to write the general formula n1 Qi = KOPi + (Pi+p + Pip) + Kni+n (11.62) 1 Using the same reasoning as before, one is led to determine the coeffi cients Kp by the system n1 KO + > 2Kp cos m E + ( 1)% = b (11.63) p=l with m assuming the values 0, 1, 2, n. Multiplying the first value by 1/2, the second by cos pu/n, the third by cos 2?, the nth by cos (n 1)p! and the last by ( )2 n n and adding them, one obtains a linear relation between the Kp, with the coefficient of Kp being (p / 0, p / n) 2 + )P+LL n ] + Cn(P ) 2 that is, n if u = p, and 0 if p. The coefficient of K0 is 2 2 2 + Cn n NACA TM 1354 The preceding conclusions remain valid, it is zero for i J 0 and equal to n if P = 0; the same result is valid for Kn. Finally, one obtains the general formula of solution (I..64) Let us consider, for instance, the operator transforming the func tion P(G) into the function dQ/de, with Q being the conjugate func tion of P; it is an operator of the second category for which bm = m Applying formula (11.64), one obtains KO m 2 n n _1 If one notes that 1n nl (x) = m cos mx 0 12 sin n 2 sin2 x 2 one sees that Kp = 0 1 n cos n if p even if p odd  1)P p / 0  x sin2 x 2) 74 NACA TM 1354 2.3.3.2 Second mode of calculation. Examination of an important particular case will show us that in certain cases it will be advantageous to consider a second mode of calculation. The method consists in replacing the function P(e) by a function of the form n D(e) = an cos nO + bn sin nO (11.66) 0 for which the method is applied with the strictest exactness; the con stants an and .bn are such that Pi = $i. One operator of the first category, one of the most important ones, is the operator of derivation which makes the function dP/de correspond to the function P(O). If we apply the first type of calculation, we shall replace ( ) by \de/i d(I ; now, it is precisely at the points 8 = in that the deriva (dei n tives and show the greatest deviation. In contrast, we shall de de obtain a good approximation of the desired function by replacing dP (2i + 1)n b d' (2i + 1)t dO 2n dO 2n We are thus led to the following mode of calculation: the circle is divided into 4n equal parts; we shall put fi = f(il) \2n/ and we shall express the 2n values Q2i as a function of the 2n value 2j+l" We shall limit ourselves to the operators of the first category. The formula expressing the Q2i as a function of P2j+l is written n >2i =_ K(P2i+2pl 2i2p+l) p=l N ACA TM 1354 and we obtain for determination of the Kp the system Ssin (2p l)m am l 2n 2 p=l with m varying from 1 to n. Multiplying the first equation by sin(2P 1), the second by 2n (20 l)2n th sin( 1)2, .,the (n 1)th 2n by sin (2P l)(n 1)n by sin the 2n ( 1 last by and adding them, one obtains a linear relation in 2 which the coefficient of Kp is n1 Z sin(2p  m=l 1)mL sin(2p  2n n1 > cos(p u)L cos(p 1 +. I )B + + C )PP "J 2 S ipn np + [L 1) + ( 1) The coefficient is zero if p 6 I, and equal to E if 2 sin (2p l)mn sin 2n ( 1)1 2 p = P. Hence (11.67) This procedure may be applied to the calculation of the derivative of a periodic function. In this case, am = m. Applying formula (II.67), one obtains )n ( 1)P+P 2n 2 NACA IM 1354 Kp= ( 1)P1 1 (1.68) [ (2p l)xl 2n 1 cos ( n) 11 2n 2..3.4 Remarks on the Employment of the Suggested Methods. In order to convey some idea of the accuracy of the proposed methods we shall give first of all a few examples where the desired results are theoretically known. Let us take as the pair of functions P(O), Q(e), the functions p(e) = 4 cos 20 4 cos 9 + 1 Q(e) 4 sin 0(2 cos 0 1) (5 4 cos 8)2 (5 4 cos 8)2 which are the real and imaginary parts, respectively, on the circle of radius 1 of the function f(z) 1 (z = ei) (2z 1)2 One will find in figure 22 the graphic representation of the func tions P(8), Q(e) and of the derivative Q'(0) of this function, and also the values of these functions for e = p (with p ranging 12 between 0 and 12). Furthermore, one will find in figure 23 the values of Q(6), calculated from P(e) as starting point, by the method just explained (coefficients Kp, defined by equation (II.61)), and in fig ure 24 on one hand the values of Q'(8), calculated from P(e) as starting point (from coefficients Kp defined by equation (II.65)), and, on the other, these same values calculated from Q(S) as starting point (coefficients Kp defined by equation (11.68)). One will see that the accuracy obtained is excellent although the selected functions show rather rapid variations. Such calculations by means of customary calculation methods are a delicate matter; this is particularly obvious in the case of the Poisson integral which is an integral "of principal value." Systematic comparisons of the method of trigonometric operators with those used so far have been made by M. Thwaites (ref. 23); they have shown that this method gives, in certain calculations, an accuracy largely superior to any attained before. The calculation procedure, with the aid of tables like the one represented (fig. 25) is very easy. One sees that one fills out this NACA IM 1354 77 table parallel to the main diagonal of the table. With such a table, about one and a half hours suffice for a Poisson integral if one has a calculating machine at his disposal. We have had occasion to point out that the accuracy of the method obviously increases to the same degree as the functions one operates with are "regular" and present "rather slight" variations. This leads in practice to two remarks which are based on the "difference method" and reasonably improve the result in certain cases. We shall, for instance, discuss the case of the Poisson integral. (1) If the function P(G) presents singularities (for instance discontinuities of the derivative for certain values of 0), it will be of interest to seek a function P1(O), presenting the same singularities as the function P(e), for which one knows explicitly the conjugate function Ql(6). One will make the calculation by means of the func tion P(O) P1(0); this function no longer presents a singularity. (2) If the function P(O) has a very extended range of variations, one will seek a function Pl(8) for which one knows explicitly the function Q%(8) so that the difference P(G) P1(0) remains of small value, and one will operate with this difference. Finally we note that, if the calculation of the derivative of a function P(e) as described above necessitates that P(O) be periodic, one can always return to this case, applying, precisely, the "difference method." 2.3.4 Example: Numerical Calculation of a Flow about a Semicircular Cone As an application, we have taken up again the case of the semicir cular cone studied in section 2.2.9. The function g(1) is given by the formula (II.48), and g'(Q) will be calculated by a Poisson inte gral. Figure 26 shows the value g'(c) thus calculated compared to the theoretical value. 25We wanted to test the accuracy of the proposed method by assuming an extremely unfavorable case, without taking into account the singu larities presented by the function g(T). For a numerical operation of great exactness, this particular case would have required application of the lemma of Schwartz, with the contour (C) completed symmetrically with respect to OX. 78 NACA TM 1354 It is then possible to calculate the representation of the pres sures, by calculating successively the function H, ZH, and the inte gral g' (p). One will find the pressure distribution thus calculated in fig ure 19; one may then compare the result obtained by the calculation method (for a very unfavorable case) with the result obtained theoretically. NACA TM 1354 CHAPTER III CONICAL FLOWS INFINITELY FLATTENED IN ONE DIRECTION The purpose of this chapter will be the study of conical flows of the second type (see chapter I, section 1.2.6). Before starting this study proper, we shall make a few remarks concerning the boundary con ditions. The conical obstacle is flattened in the direction Oxlx2. Under these conditions, reassuming the formula (1.27) x2 vx3' = (xx2' x2x3') (1 + u) (1.27) one may say that it reduces itself, in first approximation, to w2' = x2x3') (III.) since x3, x ', v, u are infinitesimals of first order, while x2 and x2' are not infinitesimals. Under these conditions, one may say that one knows the function w on the contour (C). On the other hand, one may write, within the scope of the approximations made, this boundary condition on the surface (d) of the plane Oxlx2, projection of the cone obstacle on the plane. Let us designate, provisionally, the value w by w(1)( 1x2x3) if one operates as follows (1) w(l) J1x=t),(tx(t = w(l)E1,x2Ct)J] + xj(t) xx,2(t, With the derivatives of w being, by hypothesis, supposed to be of first order, and the boundary equation written with neglect of the terms of second order, the intended simplification is justified. Various cases may arise, according to whether the cone obstacle is entirely comprised inside the Mach cone (fig. 27), whether it entirely bisects the Mach cone (fig. 28), whether the entire obstacle is com pletely outside the Mach cone (fig. 29), or whether it is partly inside and partly outside the Mach cone (fig. 30). In each of these cases there are two elementary problems, the solution of which is particularly NACA TM 1354 interesting: the first, where the relation (III.1) is reduced to w = constant = w0 which we shall call the elementary lifting problem (the corresponding flow is the flow about a delta wing placed at a certain incidence); the second, where the relation (III.1) is reduced to w = wO for x3 = +0 w = wO for x3 = 0 which we shall call the elementary symmetrical problem. This is the case of, for instance, the flow about a body consisting essentially of two delta wings, symmetrical with respect to Oxlx2 and forming an infinitely small angle with this plane. It is also the case that will be obtained, the section of which, produced by a plane parallel to Ox2xy, would be an infinitely flattened rhombus. The fact that one obtains the same mathematical formulation for two different cases indicates the relative character of the results which will be obtained. In the case of the symmetrical problem one may naturally assume that w is zero on the plane' Oxlx2 at every point situated outside of (d). Let us finally point out that very frequently the obtained results do not satisfy the conditions of linearized flows; in particular, the velocity components and their derivatives will frequently be infinite along the semiinfinite lines bounding the area (d). However, we admit once more that the results obtained provide a first approximation of the problem posed above, in accordance with the remarks made in section 1.1.3 of chapter I. 3.1 Cone Obstacle Entirely Inside the Mach Cone 3.1.1 Study of the Elementary Problems The case of the lifting cone has already formed the subject of a memorandum by Stewart (ref. 10); however, the demonstration we are going to give is more elementary and will permit us to treat simultaneously the lifting and the symmetrical case. NACA TM 1354 81 3.1.1.1 Definition of the function F(Z). We shall make our study in the plane Z. Let A'A(a,+a) be the image of the cut of the surface (d)26, (Co), as usual, the circle of radius 1 (fig. 31). Naturally, we shall operate with the function W(Z). One of the conditions to be realized which we shall find again everywhere below is that dW/dZ must be divisible by (z2 1), unless the compatibility relations show that U(Z) would admit the points Z = 1 as singular points which is inadmissible. Thus we introduce the function F(Z) Z2 dW (III.2) Z2 1 dZ iand we shall attempt to determine F(Z) for the symmetrical as well as for the lifting problem. F(Z) is a holomorphic function inside of the domain (D), bounded by the cut and the circle (CO); the only singular points this function can present on the boundary of (D), are A and A'; on the other hand, 'iF(Z) must be divisible by Z2, unless U, V, W have singularities at the origin. On the two edges of the cut F(Z) must have a real zero part. On the circle (Co) Z 1 1 Z2 1 Z 1 2i sin 0 Z Z dW = e1 dW dW dZ dZ dO Consequently, F(Z) has a real zero part on (Co) as well. The fact that F(Z) cannot be identically zero, and that its real part is zero on the boundary of (D), admits A and A' as singular points. We shall study the nature of these singularities. 3.1.1.2 Singularities of F(Z). Physically, it is clear that A Sand A' cannot be essential singular points. Let us therefore suppose .that, in the neighborhood of Z = a, one has 26One assumes, as a start, that the problem permits the use of the plane Ox1x3 as the plane of symmetry. NACA TM 1354 F(Z) ~ K%(Z a)mo m0 being arbitrary, KM / 0; let us put Z a = rei with Pq being equal to + on the upper edge of the cut, to the lower edge; for sufficiently small values of r it on orO r" O" and KmO eimo W)e must be purely imaginary quantities; thus the same will hold true for KO cos molt and for iKm sin mon; K 02 = Km02cos2m~ (i0 sin ms)2 is therefore real. On the other hand 2 sin 2mov 2 i2mus = (b cos n)( w ei sin n0g) must be real which entails sin 2m0A = 0 Thus there are two possibilities; let us denote by integral; either k an arbitrary KmO is purely imaginary or else KmO is real. mo = k + , mo = k, SNACA TM 1354 83 Let us now consider F1(Z) = F(Z) K(Z a)m0 In the neighborhood of Z = a Fl(Z) ~ Kml(Z a)ml and the same argument shows that 2mI must be an integral. Finally, one may state thefollowing theorem: Theorem: Inside of (Co) the function F(Z) may assume the form F(Z) = O(Z) + 1 t(Z) (III.3) a2 Z2 'with Q(Z) and 4(Z) admitting no singularities other than the poles at A and A'. The analysis we shall make will be simplified owing to certain symmetry conditions which F(Z) satisfies. Let us put W = w + iw' Obviously, X in w(X,Y) is even (when Y is constant). Consequently, F(Z) has a real part zero on OY. Applying Schwartz' principle one may write F(Z) = F(Z) (1I.4) This equation shows that knowledge of the development of F(Z) around Z = a immediately entails knowledge of F(Z) around Z = a. NACA TM 1354 3.1.1.3 Study of the case where F(Z) is uniform [c(Z) = O. Let us consider the function iz2P Ap(Z) = iZ2 (a2 z2)(1 a2z2 with p an integral and >1. This function satisfies all conditions imposed on F(Z). Indeed, it satisfies equation (III.4); inside of (Co) it does not admit singularities other than a and a which are poles of the order pl. Its real part is zero on the cut as well as on (CO), as one can see when writing Ap(Z) 2 (Z2 + ) a ( + \ Z2~i  (1 + a Finally, the origin should be double zero (at least). Let us assume F(Z) to be the general solution of the problem stated; we shall then demonstrate the following theorem: Theorem: If F(Z) is uniform, one has n n F(Z) = XpAp(Z) = i 11 ^a2 (III.6)  Z2)1 a2Z21 with n being an integral, and the kp being real coefficients. In case F(Z) is assumed to be a solution of the problem having a pole of the order n, one can determine a number kn so that Fl(Z) = F(Z) nAn(Z) will be a solution admitting the pole Z = a only of an order not higher than (n 1) at most. But in consequence of equation (III.4), (III.5) NACA TM 1354 F1(Z) will allow of Z = a as pole of, at most, the order (n 1). Proceeding by recurrence, one finally defines a function n Fn(Z) = F(Z) XpAp(Z) 1 which must satisfy all conditions of the problem and be holomorphic inside of (CO). The boundary conditions on the circle and on the cut entail Fn(Z) to be a constant which must be zero because Fn(Z) must become zero at the origin. 3.1.1.4 Case where O(Z) = 0. We shall study the case where O(Z) = 0 in a perfectly analogous manner. Let us put f(Z) = F(a2 z)(l a2Z2) F(Z) Z f(Z) is a uniform function inside of (Co) which admits as poles only the points (Z = a, Z = a). Actually, the origin is not a pole since, according to hypothesis, F(Z) is divisible by Z2. The function f(Z) possesses the following properties: It is imaginary on the cut, real on (Co), and real on OY (which entails properties of symmetry if one changes Z to Z). Moreover, f(Z) admits the origin as zero of, at least, the order 1. All these properties appertain equally'to the functions $r =(z 1 iZP1 2 1) Bp(Z) = Ap 1) = Z(Z2 ) p Z Z) a2 2)(1 a2Z2 p is an integral >1. Thus one deduces, as before, the theorem: Theorem: In the case where O(Z) = 0, one may write NACA TM 1354 F(Z) = i n Z2pP (III.7) z21z 2 1 a2 Z2)(1 az2] 2 with n being an integral, the Ap being real. 3.1.1.5 The principle of "minimum singularities". The for mulas (III.6) and (III.7) depend on an arbitrary number of coefficients. The only datum we know is the wo, the value w assumes on the upper edge of the cut. Thus we have to introduce a principle which will guarantee the uniqueness of the solution of the problems we have set ourselves. This principle which we shall call principle of the "minimum singularities" may be formulated in the following manner (it is con stantly being employed in mathematical physics): When the conditions of a problem require the introduction of func tions presenting singularities, one will, in a case of indeterminite ness, be satisfied with introducing the singularities of the lowest possible order permitting a complete solution of the posed problem. In the case which is of interest to us, this amounts to assuming n = 1 in the formulas (III.6) and (III.7). For the problem of interest to us, this principle has immediate significance; it amounts to stating that F(Z) and hence dW/dZ must be of an order lower than 2 in 1/Z a, or W(Z) must be of an order lower than 1 with respect to that same infinity; the considerations set forth in section 2.2.7 show that these conditions entail the total energy to remain finite. 3.1.1.6 Solution of the elementary symmetrical problem. Let us turn again to formula (III.6); one deduces from it, according to equa tion (III.2), that in the case where F(Z) is uniform dW Z2 1 ik dZ 41 (a2 Z2)(i a2Z2) and hence W(Z) = 1 log (a Z)(1 aZ) 2a(l + a2) (a + Z)(1 + aZ) NACA TM 1354 The determination of the logarithm is just that the real part of W(Z) is zero on (Co). Besides 2a(l + a2)wo On the upper edge of the cut w = w0 and on the lower edge w assumes the opposite value. This shows us that the case investigated is that of the symmetrical problem. The value W(Z) for this problem is therefore W(z) IW lo(a Z)(1 aZ) i (a + Z)(l + aZ) (III.8) The calculation of the functions U(Z) and V(Z) offers no diffi culty whatsoever. It suffices to apply the relationships of compati bility (1.25) and to integrate; the only precaution to be taken consists in choosing the constant of integration in such a manner that the real parts of U and V on (CO) become zero; one then finds v(z) ~oW (1 + a2) u(z) 2 p(1 a2 (a + Z)(1 aZ) g(Z a)(l + aZ) Z a2 S log a!  ) 1 a2Z2] This last formula is the most interesting one since it permits calcula tion of the pressure coefficient (see formula (1.8)). One finds 4C W a [loga2X2] p p 1 a2 1 a2x2 (III.9) (III.10) (III.11) 88 NACA TM 1354 In order to interpret this formula, one must connect the quanti ties a, X, to geometrical quantities, related to the given cone. Fire: of all w0 =a a being the constant hand inclination of the cone on 2X = r = tan u 1 + X2 x Ox. On the other whence = cos w 1 M2sin2 0 sin w (see fig. 32) and cos CU \ M2sin2w0 p sin a0 In figure 33 one will find the curves giving the values of functions of o, for various Mach numbers and various values of (III.12) Cp as CO' 3.1.1.7 Solution of the elementary lifting problem. If one starts from the formula (III.7), one obtains dW = dZ (2 1)2 Fa2 Z2)(1 a2z2)] 2 The integration which yields W(Z) introduces elliptic functions (see section 3.1.1.8); on the other hand, it will (now) be possible to cal culate U(Z). We note beforehand that, according to the preceding for mula, W(Z) assumes the same value on the two edges of the cut and that, consequently, this solution corresponds to the lifting problem. N:ACA TM 1354 The relationships of compatibility show that dU 2X1 dZ B z(z2 ) (a2 2) (1 a2z2)] 2 U(z) = 2X1 We s + 1)2 ka We still have to calculate 1 a Z2 + 1 Z2)(1 a2Z2 2 s a function of w, (III.13) For this purpose, one may write wO = d dZ = iil 0 dZ We put in this integral Z = iu T (z2 i) Z[ 2 1)2d [a2 2)(1 a2Z2) 2 Wo = 1 f (+ = Ull(a) S a2 + u2)( + a2u2) 2 The calculation of I(a) can be made with the aid of the function E (see ref. 24). We shall put u+ 2 u t After a few calculations one obtains I(a) =4 dt + (a t2 2 2 1)2t]2 0 1 . and hence NACA TM 1354 Finally, the change in variable sin = t(a2 + 1) 4a2 + (a2 )2t2 shows that if one puts 1 a2 1 + a2 a2(a2 + 1) O dT = 1 a2(a2 + 1) Hence the new formula for U(Z) u(z) 2 a20w a (a2 + )E1 + a2 (a2 Z2 + 1 1 _Z2)(1 a272) 2 We still have to One has (fig. 32) connect a and Z to the geometrical quantities. 2a 1 + a2  3 tan i'0 2X 1 + X2 One puts t tan u tan nLQ and obtains w0 tan l 0 1 EL 02tan2 o] t2 E a 1 + a fi^ li<*^ (III.14) = 0 tan J 